image

    Have a query? Ask a Tutor!!

    Access to best educators and exclusive paper discussions

    Can’t find what you’re looking for?

    Our tutors are from top schools around the world, and they are waiting for you 24/7, anytime, anywhere.

    • Homework- One to One Solutions
    • Understand concepts to solve better
    • Get your doubts solved instantly
    Maths-
    General
    Easy

    Question

    Polar of origin (0, 0) w.r.t. the circle x squared plus y squared plus 2 lambda x plus 2 mu y plus c equals 0 touches the circle x squared plus y squared equals r squared, if

    1. straight c equals straight r open parentheses lambda squared plus mu squared close parentheses
    2. r equals c open parentheses lambda squared plus mu squared close parentheses
    3. c squared equals r squared open parentheses lambda squared plus mu squared close parentheses
    4. r squared equals c squared open parentheses lambda squared plus mu squared close parentheses

    Hint:

    In this question, we have to find the correct option if Polar of origin (0, 0) w. r. t. the circle x squared plus y squared plus 2 lambda x plus 2 mu y plus c equals 0 touches the circle x squared plus y squared equals r squared. As we know that the polar of equation is 0. So, using this and putting P (0,0), we can find a equation for perpendicular distance of a point P open parentheses x subscript 1 comma x subscript 2 close parentheses and center (0,0) we can find the value of r and solving that we will get the correct option.

    The correct answer is: c squared equals r squared open parentheses lambda squared plus mu squared close parentheses

      E q u a t i o n space p o l o r space o f space t h e space o r i g i n space w. r. t. space t h e space c i r c l e space i s space x squared plus y squared plus 2 lambda x plus 2 mu y plus c equals 0 p o l a r space o f space e q u a t i o n space equals 0 comma space P space left parenthesis x subscript 1 comma y subscript 1 right parenthesis N o w comma space x squared plus y squared plus 2 lambda x plus 2 mu y plus c equals 0 rightwards double arrow x. x subscript 1 plus y. y subscript 1 plus 2. lambda open parentheses fraction numerator x plus x subscript 1 over denominator 2 end fraction close parentheses plus 2. mu open parentheses fraction numerator y plus y subscript 1 over denominator 2 end fraction close parentheses plus c equals 0 rightwards double arrow x. x subscript 1 plus y. y subscript 1 plus lambda left parenthesis x plus x subscript 1 right parenthesis plus mu left parenthesis y plus y subscript 1 right parenthesis plus c equals 0 rightwards double arrow x.0 plus y.0 plus lambda left parenthesis x plus 0 right parenthesis plus mu left parenthesis y plus 0 right parenthesis plus c equals 0 rightwards double arrow lambda x plus mu y plus c equals 0 G i v e n comma space x squared plus y squared equals r squared rightwards double arrow left parenthesis x minus h right parenthesis squared plus left parenthesis y minus k right parenthesis squared equals r to the power of 2 end exponent p e r p e n d i c u l a r space d i s tan c e space b e t w e e n space a space p o i n t space left parenthesis x subscript 1 comma y subscript 1 right parenthesis a n d space a space l i n e space left parenthesis lambda x plus mu y plus c equals 0 right parenthesis P equals fraction numerator open vertical bar lambda x subscript 1 plus mu y subscript 1 plus c close vertical bar over denominator square root of lambda squared plus mu squared end root end fraction rightwards double arrow r equals fraction numerator open vertical bar lambda.0 plus mu.0 plus c close vertical bar over denominator square root of lambda squared plus mu squared end root end fraction rightwards double arrow r equals fraction numerator c over denominator square root of lambda squared plus mu squared end root end fraction s q u a r i n g space b o t h space s i d e s comma rightwards double arrow r squared equals fraction numerator c squared over denominator lambda squared plus mu squared end fraction rightwards double arrow r squared open parentheses lambda squared plus mu squared close parentheses equals c squared

      Related Questions to study

      General
      physics-

      Electrons ejected from the surface of a metal, when light of certain frequency is incident on it, are stopped fully by a retarding potential of 3 volts Photo electric effect in this metallic surface begins at a frequency 6 x 1014s -1 The frequency of the incident light in s-1 is [h=6 x 10-34J-sec;charge on the electron=1.6x10-19C]

      Electrons ejected from the surface of a metal, when light of certain frequency is incident on it, are stopped fully by a retarding potential of 3 volts Photo electric effect in this metallic surface begins at a frequency 6 x 1014s -1 The frequency of the incident light in s-1 is [h=6 x 10-34J-sec;charge on the electron=1.6x10-19C]

      physics-General
      General
      physics-

      Trajectories of two projectiles are shown in figure. Let T subscript 1 end subscriptand T subscript 2 end subscript be the time periods and u subscript 1 end subscript and u subscript 2 blank end subscripttheir speeds of projection. Then

      Maximum height and time of flight depend on the vertical component of initial velocity
      H subscript 1 end subscript equals H subscript 2 end subscript rightwards double arrow u subscript y end subscript subscript 1 end subscript equals u subscript y end subscript subscript 2 end subscript
      Here T subscript 1 end subscript equals T subscript 2 end subscript
      Range R equals fraction numerator u to the power of 2 end exponent sin invisible function application 2 blank theta over denominator g end fraction equals fraction numerator 2 left parenthesis u sin invisible function application theta right parenthesis left parenthesis u cos invisible function application theta right parenthesis over denominator g end fraction
      equals fraction numerator 2 u subscript x end subscript u subscript y end subscript over denominator g end fraction
      R subscript 2 end subscript greater than R subscript 1 end subscript therefore u subscript x subscript 2 end subscript end subscript greater than u subscript x subscript 1 end subscript end subscript blank o r blank u subscript 2 end subscript greater than u subscript 1 end subscript

      Trajectories of two projectiles are shown in figure. Let T subscript 1 end subscriptand T subscript 2 end subscript be the time periods and u subscript 1 end subscript and u subscript 2 blank end subscripttheir speeds of projection. Then

      physics-General
      Maximum height and time of flight depend on the vertical component of initial velocity
      H subscript 1 end subscript equals H subscript 2 end subscript rightwards double arrow u subscript y end subscript subscript 1 end subscript equals u subscript y end subscript subscript 2 end subscript
      Here T subscript 1 end subscript equals T subscript 2 end subscript
      Range R equals fraction numerator u to the power of 2 end exponent sin invisible function application 2 blank theta over denominator g end fraction equals fraction numerator 2 left parenthesis u sin invisible function application theta right parenthesis left parenthesis u cos invisible function application theta right parenthesis over denominator g end fraction
      equals fraction numerator 2 u subscript x end subscript u subscript y end subscript over denominator g end fraction
      R subscript 2 end subscript greater than R subscript 1 end subscript therefore u subscript x subscript 2 end subscript end subscript greater than u subscript x subscript 1 end subscript end subscript blank o r blank u subscript 2 end subscript greater than u subscript 1 end subscript
      General
      physics-

      A body is projected up a smooth inclined plane with a velocity v subscript 0 end subscript from the point A as shown in figure. The angle of inclination is 45 degree and top B of the plane is connected to a well of diameter 40 m. If the body just manages to cross the well, what is the value of v subscript 0 end subscript? Length of the inclined plane is 20 square root of 2 m, and g equals 10 m s to the power of negative 2 end exponent

      Let v be the velocity acquired by the body at B which will be moving making an angle 45 degree with the horizontal direction. As the body just crosses the well so fraction numerator v to the power of 2 end exponent over denominator g end fraction equals 40
      or v to the power of 2 end exponent equals 40 g equals 40 cross times 10 equals 400
      or v equals 20 blank m s to the power of negative 1 end exponent
      Taking motion of the body from A to B along the inclined plane we have
      u equals v subscript 0 end subscript comma a equals negative g sin invisible function application 45 degree equals negative fraction numerator 10 over denominator square root of 2 end fraction m s to the power of negative 2 end exponent
      s equals 20 m comma v equals 20 m s to the power of negative 1 end exponent
      As v to the power of 2 end exponent equals u to the power of 2 end exponent plus 2 a s
      therefore blank 400 equals v subscript 0 end subscript superscript 2 end superscript plus 2 open parentheses negative fraction numerator 10 over denominator square root of 2 end fraction close parentheses cross times 20 square root of 2
      or v subscript 0 end subscript superscript 2 end superscript equals 400 plus 400 equals 800 or v equals 20 square root of 2 m s to the power of negative 1 end exponent

      A body is projected up a smooth inclined plane with a velocity v subscript 0 end subscript from the point A as shown in figure. The angle of inclination is 45 degree and top B of the plane is connected to a well of diameter 40 m. If the body just manages to cross the well, what is the value of v subscript 0 end subscript? Length of the inclined plane is 20 square root of 2 m, and g equals 10 m s to the power of negative 2 end exponent

      physics-General
      Let v be the velocity acquired by the body at B which will be moving making an angle 45 degree with the horizontal direction. As the body just crosses the well so fraction numerator v to the power of 2 end exponent over denominator g end fraction equals 40
      or v to the power of 2 end exponent equals 40 g equals 40 cross times 10 equals 400
      or v equals 20 blank m s to the power of negative 1 end exponent
      Taking motion of the body from A to B along the inclined plane we have
      u equals v subscript 0 end subscript comma a equals negative g sin invisible function application 45 degree equals negative fraction numerator 10 over denominator square root of 2 end fraction m s to the power of negative 2 end exponent
      s equals 20 m comma v equals 20 m s to the power of negative 1 end exponent
      As v to the power of 2 end exponent equals u to the power of 2 end exponent plus 2 a s
      therefore blank 400 equals v subscript 0 end subscript superscript 2 end superscript plus 2 open parentheses negative fraction numerator 10 over denominator square root of 2 end fraction close parentheses cross times 20 square root of 2
      or v subscript 0 end subscript superscript 2 end superscript equals 400 plus 400 equals 800 or v equals 20 square root of 2 m s to the power of negative 1 end exponent
      parallel
      General
      Maths-

      The equation of the circumcircle of the triangle formed by the lines y plus square root of 3 x equals 6 comma y minus square root of 3 x equals 6 and y=0 is

      L e t space t h e space t r i a n g l e space b e space increment A B C comma space w h e r e space e q u a t i o n space o f space A B space i s space y plus square root of 3 x equals 6 e q u a t i o n space o f space A C space i s space space y minus square root of 3 x equals 6 e q u a t i o n space o f space B C space i s space y equals 0 F r o m space A B thin space a n d space B C comma space w e space g e t space space space space space space space space space space space space space space space space space 0 equals 6 plus square root of 3 x rightwards double arrow negative square root of 3 x space equals 6 rightwards double arrow space space space space x space space space equals fraction numerator 6 over denominator negative square root of 3 end fraction S o comma space c o minus o r d i n a t e space o f space B space i s space open parentheses fraction numerator 6 over denominator negative square root of 3 end fraction comma 0 close parentheses. F r o m space A C thin space a n d space B C comma space w e space g e t space space space space space space space space space space space space space 0 equals 6 minus square root of 3 x rightwards double arrow square root of 3 x space equals 6 rightwards double arrow space space space space x space space space equals fraction numerator 6 over denominator square root of 3 end fraction S o comma space c o minus o r d i n a t e space o f space C space i s space open parentheses fraction numerator 6 over denominator square root of 3 end fraction comma 0 close parentheses. F r o m space A B thin space a n d space A C comma space w e space g e t space space space space space 6 plus square root of 3 x space equals 6 minus square root of 3 x rightwards double arrow space space space 2 square root of 3 x space equals 0 rightwards double arrow space space space space x space space space space space space space space space equals 0 a n d space y space equals space 6 plus square root of 3 x space equals 6 plus square root of 3 left parenthesis 0 right parenthesis equals 6 S o comma space t h e space c o minus o r d i n a t e space o f space C space i s space open parentheses 0 comma 6 close parentheses. space N o w comma space l e t space P left parenthesis h comma k right parenthesis space b e space t h e space c e n t r e space o f space t h e space c i r c l e. S o comma space A P space equals space C P rightwards double arrow open parentheses fraction numerator negative 6 over denominator square root of 3 end fraction minus h close parentheses squared plus left parenthesis 0 minus k right parenthesis squared equals open parentheses 0 minus h close parentheses squared plus open parentheses 6 minus k close parentheses squared rightwards double arrow 36 over 3 plus h squared plus fraction numerator 12 over denominator square root of 3 end fraction h plus k squared equals h squared plus 36 plus k squared minus 12 k rightwards double arrow fraction numerator 12 over denominator square root of 3 end fraction h space plus 12 k space equals space 24 rightwards double arrow fraction numerator h over denominator square root of 3 end fraction plus k equals 2 space space space space space space space space space space space space space space space space space space space space space space space space space space minus negative negative negative negative left parenthesis i right parenthesis A l s o comma space B P equals space C P rightwards double arrow open parentheses fraction numerator 6 over denominator square root of 3 end fraction minus h close parentheses squared plus open parentheses 0 minus k close parentheses squared equals open parentheses 0 minus h close parentheses squared plus open parentheses 6 minus k close parentheses squared rightwards double arrow 36 over 3 plus h squared minus fraction numerator 12 over denominator square root of 3 end fraction h plus k squared equals h squared plus 36 plus k squared minus 12 k rightwards double arrow fraction numerator negative 12 over denominator square root of 3 end fraction h space plus 12 k space equals space 24 rightwards double arrow fraction numerator negative h over denominator square root of 3 end fraction plus k equals 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus negative negative negative negative left parenthesis i i right parenthesis A d d i n g space e q u a t i o n space left parenthesis i right parenthesis space a n d space left parenthesis i i right parenthesis comma space w e space g e t colon space space space space space space fraction numerator h over denominator square root of 3 end fraction plus k minus fraction numerator h over denominator square root of 3 end fraction plus k space equals space 2 plus 2 space rightwards double arrow 2 k space equals 4 rightwards double arrow k space space space equals 2 S o comma space fraction numerator h over denominator square root of 3 end fraction plus 2 equals 2 space rightwards double arrow fraction numerator h over denominator square root of 3 end fraction space equals 0 space rightwards double arrow h space equals 0 T h u s comma space P left parenthesis h comma k right parenthesis equals P left parenthesis 0 comma 2 right parenthesis N o w comma space C P equals a equals square root of left parenthesis 0 minus 0 right parenthesis squared plus left parenthesis 6 minus 2 right parenthesis squared end root equals square root of 16 equals 4 T h e r f o r e comma space t h e space e q u a t i o n space o f space c i r c l e space rightwards double arrow left parenthesis x minus h right parenthesis squared plus left parenthesis y minus k right parenthesis squared equals a squared rightwards double arrow left parenthesis x minus 0 right parenthesis squared plus left parenthesis y minus 2 right parenthesis squared equals 16 rightwards double arrow x squared plus y squared plus 4 minus 4 y equals 16 rightwards double arrow x squared plus y squared minus 4 y equals 12

      The equation of the circumcircle of the triangle formed by the lines y plus square root of 3 x equals 6 comma y minus square root of 3 x equals 6 and y=0 is

      Maths-General
      L e t space t h e space t r i a n g l e space b e space increment A B C comma space w h e r e space e q u a t i o n space o f space A B space i s space y plus square root of 3 x equals 6 e q u a t i o n space o f space A C space i s space space y minus square root of 3 x equals 6 e q u a t i o n space o f space B C space i s space y equals 0 F r o m space A B thin space a n d space B C comma space w e space g e t space space space space space space space space space space space space space space space space space 0 equals 6 plus square root of 3 x rightwards double arrow negative square root of 3 x space equals 6 rightwards double arrow space space space space x space space space equals fraction numerator 6 over denominator negative square root of 3 end fraction S o comma space c o minus o r d i n a t e space o f space B space i s space open parentheses fraction numerator 6 over denominator negative square root of 3 end fraction comma 0 close parentheses. F r o m space A C thin space a n d space B C comma space w e space g e t space space space space space space space space space space space space space 0 equals 6 minus square root of 3 x rightwards double arrow square root of 3 x space equals 6 rightwards double arrow space space space space x space space space equals fraction numerator 6 over denominator square root of 3 end fraction S o comma space c o minus o r d i n a t e space o f space C space i s space open parentheses fraction numerator 6 over denominator square root of 3 end fraction comma 0 close parentheses. F r o m space A B thin space a n d space A C comma space w e space g e t space space space space space 6 plus square root of 3 x space equals 6 minus square root of 3 x rightwards double arrow space space space 2 square root of 3 x space equals 0 rightwards double arrow space space space space x space space space space space space space space space equals 0 a n d space y space equals space 6 plus square root of 3 x space equals 6 plus square root of 3 left parenthesis 0 right parenthesis equals 6 S o comma space t h e space c o minus o r d i n a t e space o f space C space i s space open parentheses 0 comma 6 close parentheses. space N o w comma space l e t space P left parenthesis h comma k right parenthesis space b e space t h e space c e n t r e space o f space t h e space c i r c l e. S o comma space A P space equals space C P rightwards double arrow open parentheses fraction numerator negative 6 over denominator square root of 3 end fraction minus h close parentheses squared plus left parenthesis 0 minus k right parenthesis squared equals open parentheses 0 minus h close parentheses squared plus open parentheses 6 minus k close parentheses squared rightwards double arrow 36 over 3 plus h squared plus fraction numerator 12 over denominator square root of 3 end fraction h plus k squared equals h squared plus 36 plus k squared minus 12 k rightwards double arrow fraction numerator 12 over denominator square root of 3 end fraction h space plus 12 k space equals space 24 rightwards double arrow fraction numerator h over denominator square root of 3 end fraction plus k equals 2 space space space space space space space space space space space space space space space space space space space space space space space space space space minus negative negative negative negative left parenthesis i right parenthesis A l s o comma space B P equals space C P rightwards double arrow open parentheses fraction numerator 6 over denominator square root of 3 end fraction minus h close parentheses squared plus open parentheses 0 minus k close parentheses squared equals open parentheses 0 minus h close parentheses squared plus open parentheses 6 minus k close parentheses squared rightwards double arrow 36 over 3 plus h squared minus fraction numerator 12 over denominator square root of 3 end fraction h plus k squared equals h squared plus 36 plus k squared minus 12 k rightwards double arrow fraction numerator negative 12 over denominator square root of 3 end fraction h space plus 12 k space equals space 24 rightwards double arrow fraction numerator negative h over denominator square root of 3 end fraction plus k equals 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus negative negative negative negative left parenthesis i i right parenthesis A d d i n g space e q u a t i o n space left parenthesis i right parenthesis space a n d space left parenthesis i i right parenthesis comma space w e space g e t colon space space space space space space fraction numerator h over denominator square root of 3 end fraction plus k minus fraction numerator h over denominator square root of 3 end fraction plus k space equals space 2 plus 2 space rightwards double arrow 2 k space equals 4 rightwards double arrow k space space space equals 2 S o comma space fraction numerator h over denominator square root of 3 end fraction plus 2 equals 2 space rightwards double arrow fraction numerator h over denominator square root of 3 end fraction space equals 0 space rightwards double arrow h space equals 0 T h u s comma space P left parenthesis h comma k right parenthesis equals P left parenthesis 0 comma 2 right parenthesis N o w comma space C P equals a equals square root of left parenthesis 0 minus 0 right parenthesis squared plus left parenthesis 6 minus 2 right parenthesis squared end root equals square root of 16 equals 4 T h e r f o r e comma space t h e space e q u a t i o n space o f space c i r c l e space rightwards double arrow left parenthesis x minus h right parenthesis squared plus left parenthesis y minus k right parenthesis squared equals a squared rightwards double arrow left parenthesis x minus 0 right parenthesis squared plus left parenthesis y minus 2 right parenthesis squared equals 16 rightwards double arrow x squared plus y squared plus 4 minus 4 y equals 16 rightwards double arrow x squared plus y squared minus 4 y equals 12
      General
      physics-

      Which of the substance A comma blank B or C has the highest specific heat? The temperature v s time graph is shown

      Substances having more specific heat take longer time to get heated to a higher temperature and longer time to get cooled.

      If we draw a line parallel to the time axis then it cuts the given graphs at three different points. Corresponding points on the times axis shows that
      t subscript C end subscript greater than t subscript B end subscript greater than t subscript A end subscript rightwards double arrow C subscript C end subscript greater than C subscript B end subscript greater than C subscript A end subscript

      Which of the substance A comma blank B or C has the highest specific heat? The temperature v s time graph is shown

      physics-General
      Substances having more specific heat take longer time to get heated to a higher temperature and longer time to get cooled.

      If we draw a line parallel to the time axis then it cuts the given graphs at three different points. Corresponding points on the times axis shows that
      t subscript C end subscript greater than t subscript B end subscript greater than t subscript A end subscript rightwards double arrow C subscript C end subscript greater than C subscript B end subscript greater than C subscript A end subscript
      General
      physics-

      The figure shows a system of two concentric spheres of radii r subscript 1 end subscript and r subscript 2 end subscript and kept at temperatures T subscript 1 end subscript and T subscript 2 end subscript respectively. The radial rate of flow of heat in a substance between the two concentric spheres, is proportional to

      To measure the radial rate of heat flow, we have to go for integration technique as here the area of the surface through which heat will flow is not constant.

      Let us consider an element (spherical shell) of thickness dx and radius x as shown in figure. Let us first find the equivalent thermal resistance between inner and outer sphere.
      Resistance of shell=d R equals blank fraction numerator d x over denominator K cross times 4 pi x to the power of 2 end exponent end fraction
      open parentheses fraction numerator F r o m blank R equals fraction numerator l over denominator K A end fraction w h e r e comma over denominator K equals t h e r m a l blank c o n d u c t i v i t y end fraction close parentheses
      rightwards double arrow not stretchy integral d R equals R equals not stretchy integral subscript r subscript 1 end subscript end subscript superscript r subscript 2 end subscript end superscript fraction numerator d x over denominator 4 pi K x to the power of 2 blank end exponent end fraction
      = fraction numerator 1 over denominator 4 pi K end fraction open square brackets fraction numerator 1 over denominator r subscript 1 end subscript end fraction minus blank fraction numerator 1 over denominator r subscript 2 end subscript end fraction close square brackets equals blank fraction numerator r subscript 2 end subscript minus r subscript 1 end subscript over denominator 4 pi K left parenthesis r subscript 1 end subscript r subscript 2 end subscript right parenthesis end fraction
      Rate of heat flow = H
      = fraction numerator T subscript 1 end subscript minus T subscript 2 end subscript over denominator R end fraction
      = fraction numerator T subscript 1 end subscript minus T subscript 2 end subscript over denominator r subscript 2 end subscript minus r subscript 1 end subscript end fraction blank cross times 4 pi K open parentheses r subscript 1 end subscript r subscript 2 end subscript close parentheses
      proportional to fraction numerator r subscript 1 end subscript r subscript 2 end subscript over denominator r subscript 2 end subscript minus blank r subscript 1 end subscript end fraction

      The figure shows a system of two concentric spheres of radii r subscript 1 end subscript and r subscript 2 end subscript and kept at temperatures T subscript 1 end subscript and T subscript 2 end subscript respectively. The radial rate of flow of heat in a substance between the two concentric spheres, is proportional to

      physics-General
      To measure the radial rate of heat flow, we have to go for integration technique as here the area of the surface through which heat will flow is not constant.

      Let us consider an element (spherical shell) of thickness dx and radius x as shown in figure. Let us first find the equivalent thermal resistance between inner and outer sphere.
      Resistance of shell=d R equals blank fraction numerator d x over denominator K cross times 4 pi x to the power of 2 end exponent end fraction
      open parentheses fraction numerator F r o m blank R equals fraction numerator l over denominator K A end fraction w h e r e comma over denominator K equals t h e r m a l blank c o n d u c t i v i t y end fraction close parentheses
      rightwards double arrow not stretchy integral d R equals R equals not stretchy integral subscript r subscript 1 end subscript end subscript superscript r subscript 2 end subscript end superscript fraction numerator d x over denominator 4 pi K x to the power of 2 blank end exponent end fraction
      = fraction numerator 1 over denominator 4 pi K end fraction open square brackets fraction numerator 1 over denominator r subscript 1 end subscript end fraction minus blank fraction numerator 1 over denominator r subscript 2 end subscript end fraction close square brackets equals blank fraction numerator r subscript 2 end subscript minus r subscript 1 end subscript over denominator 4 pi K left parenthesis r subscript 1 end subscript r subscript 2 end subscript right parenthesis end fraction
      Rate of heat flow = H
      = fraction numerator T subscript 1 end subscript minus T subscript 2 end subscript over denominator R end fraction
      = fraction numerator T subscript 1 end subscript minus T subscript 2 end subscript over denominator r subscript 2 end subscript minus r subscript 1 end subscript end fraction blank cross times 4 pi K open parentheses r subscript 1 end subscript r subscript 2 end subscript close parentheses
      proportional to fraction numerator r subscript 1 end subscript r subscript 2 end subscript over denominator r subscript 2 end subscript minus blank r subscript 1 end subscript end fraction
      parallel
      General
      maths-

      If the line y = x + 3 meets the circle x squared plus y squared equals a squared at A and B, then equation of the circle on AB as diameter is

      If the line y = x + 3 meets the circle x squared plus y squared equals a squared at A and B, then equation of the circle on AB as diameter is

      maths-General
      General
      physics-

      An electric lamp is fixed at the ceiling of a circular tunnel as shown is figure. What is the ratio the intensities of light at base A and a point B on the wall

      I subscript A end subscript equals fraction numerator L over denominator left parenthesis 2 r right parenthesis to the power of 2 end exponent end fraction and I subscript B end subscript equals fraction numerator L over denominator open parentheses r square root of 2 close parentheses to the power of 2 end exponent end fraction cos invisible function application theta
      equals fraction numerator L over denominator 2 r to the power of 2 end exponent end fraction. fraction numerator r over denominator r square root of 2 end fraction equals fraction numerator L over denominator 2 square root of 2   r to the power of 2 end exponent end fraction

      therefore fraction numerator I subscript A end subscript over denominator I subscript B end subscript end fraction equals fraction numerator 2 square root of 2 over denominator 4 end fraction equals fraction numerator 1 over denominator square root of 2 end fraction

      An electric lamp is fixed at the ceiling of a circular tunnel as shown is figure. What is the ratio the intensities of light at base A and a point B on the wall

      physics-General
      I subscript A end subscript equals fraction numerator L over denominator left parenthesis 2 r right parenthesis to the power of 2 end exponent end fraction and I subscript B end subscript equals fraction numerator L over denominator open parentheses r square root of 2 close parentheses to the power of 2 end exponent end fraction cos invisible function application theta
      equals fraction numerator L over denominator 2 r to the power of 2 end exponent end fraction. fraction numerator r over denominator r square root of 2 end fraction equals fraction numerator L over denominator 2 square root of 2   r to the power of 2 end exponent end fraction

      therefore fraction numerator I subscript A end subscript over denominator I subscript B end subscript end fraction equals fraction numerator 2 square root of 2 over denominator 4 end fraction equals fraction numerator 1 over denominator square root of 2 end fraction
      General
      physics-

      The distance between a point source of light and a screen which is 60 cm is increased to 180 cm. The intensity on the screen as compared with the original intensity will be

      I proportional to fraction numerator 1 over denominator r to the power of 2 end exponent end fraction rightwards double arrow fraction numerator I subscript 2 end subscript over denominator I subscript 1 end subscript end fraction equals fraction numerator r subscript 1 end subscript superscript 2 end superscript over denominator r subscript 2 end subscript superscript 2 end superscript end fraction equals fraction numerator 6 0 to the power of 2 end exponent over denominator 18 0 to the power of 2 end exponent end fraction equals fraction numerator 1 over denominator 9 end fraction

      The distance between a point source of light and a screen which is 60 cm is increased to 180 cm. The intensity on the screen as compared with the original intensity will be

      physics-General
      I proportional to fraction numerator 1 over denominator r to the power of 2 end exponent end fraction rightwards double arrow fraction numerator I subscript 2 end subscript over denominator I subscript 1 end subscript end fraction equals fraction numerator r subscript 1 end subscript superscript 2 end superscript over denominator r subscript 2 end subscript superscript 2 end superscript end fraction equals fraction numerator 6 0 to the power of 2 end exponent over denominator 18 0 to the power of 2 end exponent end fraction equals fraction numerator 1 over denominator 9 end fraction
      parallel
      General
      Maths-

      The pole of the straight line 9x + y – 28 = 0 with respect to the circle 2 x squared plus 2 y squared minus 3 x plus 5 y minus 7 equals 0 is

      Let the coordinates of the pole be (h, k).
      The equation of the circle is 2x2 + 2y2 - 3x + 5y - 7 = 0.
      Dividing both sides of the equation by 2,we get
      x2 + y2 - 3/2 x + 5/2 y - 7/2 = 0.
      The equation of polar for the point having coordinates (h, k) is
      h x + k y - 3/4 (x + h) + 5/4 (y+k) - 7/2 = 0 .
      Simplifying the equation, it can be written as (4h - 3)x + (4k + 5)y + (-3h + 5k - 14).
      We are given that the equation of the polar is 9x + y - 28 = 0.so, the above equation and this equation represent the same line. Comparing the coefficient of the line, we get
      (4h - 3) /9 = (4k + 5) /1 = (-3h + 5k - 14) /-28
      Taking the first and third expression, we get
      (4h - 3) /9 = (-3h + 5k - 14) /-28 .
      By cross multiplication, we get
      Or, -28 (4h - 3) = 9 (-3h + 5k - 14)
      Or, -112h + 84 = -27h + 45k - 126
      Or, -112h + 27h - 45k = -126 - 84
      Or, -85h - 45k = -210
      Or, -5 ( 17h + 9k ) = -210
      17h + 9k = 42 . . . (1)
      Taking the second and third expression, we get
      (4k + 5) /1 = ( -3h + 5k - 14) /-28
      By cross multiplication, we get
      -28 ( 4k + 5 ) = ( -3h + 5k - 14 )
      -112k - 140 = -3h + 5k - 14
      3h - 112k - 5k = -14 + 140
      3h - 117k = 126 . . . (2)
      By solving equation (1) × 13 and equation (2) .
      h = 3
      By putting the value of h in equation (1) , we get
      K = -1 .
      Therefore, the coordinates are (3, -1) .
      Hence, the correct option is (a).

      The pole of the straight line 9x + y – 28 = 0 with respect to the circle 2 x squared plus 2 y squared minus 3 x plus 5 y minus 7 equals 0 is

      Maths-General
      Let the coordinates of the pole be (h, k).
      The equation of the circle is 2x2 + 2y2 - 3x + 5y - 7 = 0.
      Dividing both sides of the equation by 2,we get
      x2 + y2 - 3/2 x + 5/2 y - 7/2 = 0.
      The equation of polar for the point having coordinates (h, k) is
      h x + k y - 3/4 (x + h) + 5/4 (y+k) - 7/2 = 0 .
      Simplifying the equation, it can be written as (4h - 3)x + (4k + 5)y + (-3h + 5k - 14).
      We are given that the equation of the polar is 9x + y - 28 = 0.so, the above equation and this equation represent the same line. Comparing the coefficient of the line, we get
      (4h - 3) /9 = (4k + 5) /1 = (-3h + 5k - 14) /-28
      Taking the first and third expression, we get
      (4h - 3) /9 = (-3h + 5k - 14) /-28 .
      By cross multiplication, we get
      Or, -28 (4h - 3) = 9 (-3h + 5k - 14)
      Or, -112h + 84 = -27h + 45k - 126
      Or, -112h + 27h - 45k = -126 - 84
      Or, -85h - 45k = -210
      Or, -5 ( 17h + 9k ) = -210
      17h + 9k = 42 . . . (1)
      Taking the second and third expression, we get
      (4k + 5) /1 = ( -3h + 5k - 14) /-28
      By cross multiplication, we get
      -28 ( 4k + 5 ) = ( -3h + 5k - 14 )
      -112k - 140 = -3h + 5k - 14
      3h - 112k - 5k = -14 + 140
      3h - 117k = 126 . . . (2)
      By solving equation (1) × 13 and equation (2) .
      h = 3
      By putting the value of h in equation (1) , we get
      K = -1 .
      Therefore, the coordinates are (3, -1) .
      Hence, the correct option is (a).
      General
      physics-

      Three rods of equal length l are joined to form an equilateral triangle P Q R. O is the mid point of P Q. Distance O R remains same for small change in temperature. Coefficient of linear expansion for P R and R Q is same, i. e. comma blank alpha subscript 2 end subscript but that for P Q is alpha subscript 1 end subscript. Then

      open parentheses O R close parentheses to the power of 2 end exponent equals open parentheses P R close parentheses to the power of 2 end exponent minus open parentheses P O close parentheses to the power of 2 end exponent equals l to the power of 2 end exponent minus open parentheses fraction numerator l over denominator 2 end fraction close parentheses to the power of 2 end exponent
      equals open square brackets l open parentheses 1 plus alpha subscript 2 end subscript t close parentheses close square brackets to the power of 2 end exponent minus open square brackets fraction numerator l over denominator 2 end fraction open parentheses 1 plus alpha subscript 1 end subscript t close parentheses close square brackets to the power of 2 end exponentl to the power of 2 end exponent minus fraction numerator l to the power of 2 end exponent over denominator 4 end fraction equals l to the power of 2 end exponent open parentheses 1 plus alpha subscript 2 end subscript superscript 2 end superscript t to the power of 2 end exponent plus 2 alpha subscript 2 end subscript t close parentheses minus fraction numerator l to the power of 2 end exponent over denominator 4 end fraction left parenthesis 1 plus alpha subscript 1 end subscript superscript 2 end superscript t to the power of 2 end exponent plus 2 alpha subscript 1 end subscript t right parenthesis
      Neglecting alpha subscript 2 end subscript superscript 2 end superscript t to the power of 2 end exponent and alpha subscript 1 end subscript superscript 2 end superscript t to the power of 2 end exponent
      0 equals l to the power of 2 end exponent open parentheses 2 alpha subscript 2 end subscript t close parentheses minus fraction numerator l to the power of 2 end exponent over denominator 4 end fraction open parentheses 2 alpha subscript 1 end subscript t close parentheses rightwards double arrow 2 alpha subscript 2 end subscript equals fraction numerator 2 alpha subscript 1 end subscript over denominator 4 end fraction rightwards double arrow alpha subscript 1 end subscript equals 4 alpha subscript 2 end subscript

      Three rods of equal length l are joined to form an equilateral triangle P Q R. O is the mid point of P Q. Distance O R remains same for small change in temperature. Coefficient of linear expansion for P R and R Q is same, i. e. comma blank alpha subscript 2 end subscript but that for P Q is alpha subscript 1 end subscript. Then

      physics-General
      open parentheses O R close parentheses to the power of 2 end exponent equals open parentheses P R close parentheses to the power of 2 end exponent minus open parentheses P O close parentheses to the power of 2 end exponent equals l to the power of 2 end exponent minus open parentheses fraction numerator l over denominator 2 end fraction close parentheses to the power of 2 end exponent
      equals open square brackets l open parentheses 1 plus alpha subscript 2 end subscript t close parentheses close square brackets to the power of 2 end exponent minus open square brackets fraction numerator l over denominator 2 end fraction open parentheses 1 plus alpha subscript 1 end subscript t close parentheses close square brackets to the power of 2 end exponentl to the power of 2 end exponent minus fraction numerator l to the power of 2 end exponent over denominator 4 end fraction equals l to the power of 2 end exponent open parentheses 1 plus alpha subscript 2 end subscript superscript 2 end superscript t to the power of 2 end exponent plus 2 alpha subscript 2 end subscript t close parentheses minus fraction numerator l to the power of 2 end exponent over denominator 4 end fraction left parenthesis 1 plus alpha subscript 1 end subscript superscript 2 end superscript t to the power of 2 end exponent plus 2 alpha subscript 1 end subscript t right parenthesis
      Neglecting alpha subscript 2 end subscript superscript 2 end superscript t to the power of 2 end exponent and alpha subscript 1 end subscript superscript 2 end superscript t to the power of 2 end exponent
      0 equals l to the power of 2 end exponent open parentheses 2 alpha subscript 2 end subscript t close parentheses minus fraction numerator l to the power of 2 end exponent over denominator 4 end fraction open parentheses 2 alpha subscript 1 end subscript t close parentheses rightwards double arrow 2 alpha subscript 2 end subscript equals fraction numerator 2 alpha subscript 1 end subscript over denominator 4 end fraction rightwards double arrow alpha subscript 1 end subscript equals 4 alpha subscript 2 end subscript
      General
      physics-

      A point source causes photoelectric effect from a small metal plate Which of the following curves may represent the saturation photocurrent as a function of the distance between the source and the metal?

      A point source causes photoelectric effect from a small metal plate Which of the following curves may represent the saturation photocurrent as a function of the distance between the source and the metal?

      physics-General
      parallel
      General
      physics-

      One of the following figures respesents the variation of particle momentum with associated de Broglie wavelength
      a)
      b)
      c)

      One of the following figures respesents the variation of particle momentum with associated de Broglie wavelength
      a)
      b)
      c)

      physics-General
      General
      physics-

      Two circular discs A and B with equal radii are blackened. They are heated to some temperature and are cooled under identical conditions. What inference do you draw from their cooling curves?

      According to Newton’s law of cooling, rate of cooling is given by
      open parentheses fraction numerator negative d T over denominator d t end fraction close parentheses equals fraction numerator e A sigma over denominator m c end fraction left parenthesis T to the power of 4 end exponent minus T subscript 0 end subscript superscript 4 end superscript right parenthesis
      Where c is specific heat of material.
      or open parentheses fraction numerator negative d T over denominator d t end fraction close parentheses proportional to fraction numerator 1 over denominator c end fraction
      i e. comma blankrate of cooling varies inversely as specific heat. From the graph, for A rate of cooling is larger. Therefore, specific heat of A is smaller.

      Two circular discs A and B with equal radii are blackened. They are heated to some temperature and are cooled under identical conditions. What inference do you draw from their cooling curves?

      physics-General
      According to Newton’s law of cooling, rate of cooling is given by
      open parentheses fraction numerator negative d T over denominator d t end fraction close parentheses equals fraction numerator e A sigma over denominator m c end fraction left parenthesis T to the power of 4 end exponent minus T subscript 0 end subscript superscript 4 end superscript right parenthesis
      Where c is specific heat of material.
      or open parentheses fraction numerator negative d T over denominator d t end fraction close parentheses proportional to fraction numerator 1 over denominator c end fraction
      i e. comma blankrate of cooling varies inversely as specific heat. From the graph, for A rate of cooling is larger. Therefore, specific heat of A is smaller.
      General
      physics-

      Which of the curves in figure represents the relation between Celsius and Fahrenheit temperatures

      fraction numerator C over denominator 5 end fraction equals fraction numerator F minus 32 over denominator 9 end fraction rightwards double arrow C equals open parentheses fraction numerator 5 over denominator 9 end fraction close parentheses F minus fraction numerator 20 over denominator 3 end fraction. Hence graph between ℃ and ℉ will be a straight line with positive slope and negative intercept

      Which of the curves in figure represents the relation between Celsius and Fahrenheit temperatures

      physics-General
      fraction numerator C over denominator 5 end fraction equals fraction numerator F minus 32 over denominator 9 end fraction rightwards double arrow C equals open parentheses fraction numerator 5 over denominator 9 end fraction close parentheses F minus fraction numerator 20 over denominator 3 end fraction. Hence graph between ℃ and ℉ will be a straight line with positive slope and negative intercept
      parallel

      card img

      With Turito Academy.

      card img

      With Turito Foundation.

      card img

      Get an Expert Advice From Turito.

      Turito Academy

      card img

      With Turito Academy.

      Test Prep

      card img

      With Turito Foundation.