Physics-
General
Easy

Question

The figures represent three cases of a ray passing through a prism of angle A. The case corresponding to minimum deviation is

  1. 1    
  2. 2    
  3. 3    
  4. None of these    

The correct answer is: 3

Related Questions to study

General
Maths-

If the normal at the point straight P left parenthesis theta right parenthesis to the ellipse x squared over 14 plus y squared over 5 equals 1 intersects it again at the point straight Q left parenthesis 2 theta right parenthesis, then cos space theta is

G i v e n comma space t h e space e q u a t i o n space o f space e l l i p s e space x squared over 14 plus y squared over 5 equals 1
O n space c o m p a r i n g space w i t h space s tan d a r d space e q u a t i o n space o f space e l l i p s e space x squared over a squared plus y squared over b squared equals 1
a squared equals 14 space a n d space b squared equals 5
w e space k n o w space t h a t comma space p o i n t space o n space a n space e l l i p s e space i n space t h e space p a r a m e t r i c space f o r m space i s space left parenthesis a cos space x comma space b space sin space x right parenthesis
S o comma space P open parentheses theta close parentheses equals left parenthesis a cos theta comma space b sin theta right parenthesis
space space space space space space space space Q left parenthesis 2 theta right parenthesis equals left parenthesis a cos 2 theta comma space b sin 2 theta right parenthesis
s l o p e space o f space l i n e space P Q space equals fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction equals fraction numerator b sin 2 theta minus b sin theta over denominator a cos 2 theta minus a cos theta end fraction
A l s o comma space s l o p e space o f space n o r m a l space equals space a over b tan theta
A s space t h e space n o r a m l space p a s s e s space t h r o u g h space P Q comma
fraction numerator b sin 2 theta minus b sin theta over denominator a cos 2 theta minus a cos theta end fraction equals a over b tan theta
fraction numerator b left parenthesis sin 2 theta minus sin theta right parenthesis over denominator a left parenthesis cos 2 theta minus cos theta right parenthesis end fraction equals a over b fraction numerator sin theta over denominator cos theta end fraction 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 open square brackets tan theta equals fraction numerator sin theta over denominator cos theta end fraction close square brackets
fraction numerator b left parenthesis 2 sin theta cos theta minus sin theta right parenthesis over denominator a left parenthesis 2 cos squared theta minus 1 minus cos theta right parenthesis end fraction equals a over b fraction numerator sin theta over denominator cos theta end fraction space space space space space space space space space space space space space space space space open square brackets sin 2 theta equals 2 sin theta. cos theta space a n d space cos 2 theta equals 2 cos squared theta minus 1 close square brackets
fraction numerator b sin theta left parenthesis 2 cos theta minus 1 right parenthesis over denominator a left parenthesis 2 cos squared theta minus 1 minus cos theta right parenthesis end fraction equals a over b fraction numerator sin theta over denominator cos theta end fraction
fraction numerator b left parenthesis 2 cos theta minus 1 right parenthesis over denominator a left parenthesis 2 cos squared theta minus 1 minus cos theta right parenthesis end fraction equals a over b fraction numerator 1 over denominator cos theta end fraction
fraction numerator b squared left parenthesis 2 cos theta minus 1 right parenthesis over denominator a squared left parenthesis 2 cos squared theta minus 1 minus cos theta right parenthesis end fraction equals fraction numerator 1 over denominator cos theta end fraction
fraction numerator 5 left parenthesis 2 cos theta minus 1 right parenthesis over denominator 14 left parenthesis 2 cos squared theta minus 1 minus cos theta right parenthesis end fraction equals fraction numerator 1 over denominator cos theta end fraction
5 left parenthesis 2 cos theta minus 1 right parenthesis cos theta equals 14 left parenthesis 2 cos squared theta minus 1 minus cos theta right parenthesis
10 cos squared theta minus 5 cos theta equals 28 cos squared theta minus 14 minus 14 cos theta
28 cos squared theta minus 10 cos squared theta minus 14 cos theta plus cos theta minus 14 equals 0
18 cos squared theta minus 9 cos theta minus 14 equals 0
18 cos squared theta minus 21 cos theta plus 12 cos theta minus 14 equals 0
3 cos theta left parenthesis 6 cos theta minus 7 right parenthesis plus 2 left parenthesis 6 cos theta minus 7 right parenthesis equals 0
left parenthesis 3 cos theta plus 2 right parenthesis left parenthesis 6 cos theta minus 7 right parenthesis equals 0
i f comma left parenthesis 6 cos theta minus 7 right parenthesis equals 0
cos theta equals 7 over 6 space n o t space p o s s i b l e space a s space v a l u e space a c n space n o t space b e space g r e a t e r space t h a n space 1.
i f left parenthesis 3 cos theta plus 2 right parenthesis equals 0
cos theta equals fraction numerator negative 2 over denominator 3 end fraction
S o comma space t h e space r e q u i r e d space a n s w e r space i s space fraction numerator negative 2 over denominator 3 end fraction.

 

If the normal at the point straight P left parenthesis theta right parenthesis to the ellipse x squared over 14 plus y squared over 5 equals 1 intersects it again at the point straight Q left parenthesis 2 theta right parenthesis, then cos space theta is

Maths-General
G i v e n comma space t h e space e q u a t i o n space o f space e l l i p s e space x squared over 14 plus y squared over 5 equals 1
O n space c o m p a r i n g space w i t h space s tan d a r d space e q u a t i o n space o f space e l l i p s e space x squared over a squared plus y squared over b squared equals 1
a squared equals 14 space a n d space b squared equals 5
w e space k n o w space t h a t comma space p o i n t space o n space a n space e l l i p s e space i n space t h e space p a r a m e t r i c space f o r m space i s space left parenthesis a cos space x comma space b space sin space x right parenthesis
S o comma space P open parentheses theta close parentheses equals left parenthesis a cos theta comma space b sin theta right parenthesis
space space space space space space space space Q left parenthesis 2 theta right parenthesis equals left parenthesis a cos 2 theta comma space b sin 2 theta right parenthesis
s l o p e space o f space l i n e space P Q space equals fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction equals fraction numerator b sin 2 theta minus b sin theta over denominator a cos 2 theta minus a cos theta end fraction
A l s o comma space s l o p e space o f space n o r m a l space equals space a over b tan theta
A s space t h e space n o r a m l space p a s s e s space t h r o u g h space P Q comma
fraction numerator b sin 2 theta minus b sin theta over denominator a cos 2 theta minus a cos theta end fraction equals a over b tan theta
fraction numerator b left parenthesis sin 2 theta minus sin theta right parenthesis over denominator a left parenthesis cos 2 theta minus cos theta right parenthesis end fraction equals a over b fraction numerator sin theta over denominator cos theta end fraction 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 open square brackets tan theta equals fraction numerator sin theta over denominator cos theta end fraction close square brackets
fraction numerator b left parenthesis 2 sin theta cos theta minus sin theta right parenthesis over denominator a left parenthesis 2 cos squared theta minus 1 minus cos theta right parenthesis end fraction equals a over b fraction numerator sin theta over denominator cos theta end fraction space space space space space space space space space space space space space space space space open square brackets sin 2 theta equals 2 sin theta. cos theta space a n d space cos 2 theta equals 2 cos squared theta minus 1 close square brackets
fraction numerator b sin theta left parenthesis 2 cos theta minus 1 right parenthesis over denominator a left parenthesis 2 cos squared theta minus 1 minus cos theta right parenthesis end fraction equals a over b fraction numerator sin theta over denominator cos theta end fraction
fraction numerator b left parenthesis 2 cos theta minus 1 right parenthesis over denominator a left parenthesis 2 cos squared theta minus 1 minus cos theta right parenthesis end fraction equals a over b fraction numerator 1 over denominator cos theta end fraction
fraction numerator b squared left parenthesis 2 cos theta minus 1 right parenthesis over denominator a squared left parenthesis 2 cos squared theta minus 1 minus cos theta right parenthesis end fraction equals fraction numerator 1 over denominator cos theta end fraction
fraction numerator 5 left parenthesis 2 cos theta minus 1 right parenthesis over denominator 14 left parenthesis 2 cos squared theta minus 1 minus cos theta right parenthesis end fraction equals fraction numerator 1 over denominator cos theta end fraction
5 left parenthesis 2 cos theta minus 1 right parenthesis cos theta equals 14 left parenthesis 2 cos squared theta minus 1 minus cos theta right parenthesis
10 cos squared theta minus 5 cos theta equals 28 cos squared theta minus 14 minus 14 cos theta
28 cos squared theta minus 10 cos squared theta minus 14 cos theta plus cos theta minus 14 equals 0
18 cos squared theta minus 9 cos theta minus 14 equals 0
18 cos squared theta minus 21 cos theta plus 12 cos theta minus 14 equals 0
3 cos theta left parenthesis 6 cos theta minus 7 right parenthesis plus 2 left parenthesis 6 cos theta minus 7 right parenthesis equals 0
left parenthesis 3 cos theta plus 2 right parenthesis left parenthesis 6 cos theta minus 7 right parenthesis equals 0
i f comma left parenthesis 6 cos theta minus 7 right parenthesis equals 0
cos theta equals 7 over 6 space n o t space p o s s i b l e space a s space v a l u e space a c n space n o t space b e space g r e a t e r space t h a n space 1.
i f left parenthesis 3 cos theta plus 2 right parenthesis equals 0
cos theta equals fraction numerator negative 2 over denominator 3 end fraction
S o comma space t h e space r e q u i r e d space a n s w e r space i s space fraction numerator negative 2 over denominator 3 end fraction.

 
General
physics-

A fighter plane enters inside the enemy territory, at time t equals 0 with velocity v subscript 0 end subscript equals 250 blank m s to the power of negative 1 end exponent and moves horizontally with constant acceleration a equals 20 m s to the power of negative 2 end exponent (see figure). An enemy tank at the border, spot the plane and fire shots at an angle theta equals 60 degree with the horizontal and with velocity u equals 600 blank m s to the power of negative 1 end exponent. At what altitude H of the plane it can be hit by the shot?

If it is being hit then
d equals v subscript 0 end subscript t plus fraction numerator 1 over denominator 2 end fraction a t to the power of 2 end exponent equals left parenthesis u cos invisible function application theta right parenthesis t
or t equals fraction numerator u cos invisible function application theta minus v subscript 0 end subscript over denominator a divided by 2 end fraction

therefore blank t equals fraction numerator 600 cross times fraction numerator 1 over denominator 2 end fraction minus 250 over denominator 10 end fraction equals 5 blank s
H equals open parentheses u sin invisible function application theta close parentheses t minus fraction numerator 1 over denominator 2 end fraction cross times g t to the power of 2 end exponent
equals 600 cross times fraction numerator square root of 3 over denominator 2 end fraction cross times 5 minus fraction numerator 1 over denominator 2 end fraction cross times 10 cross times 25
H equals 2473 blank m

A fighter plane enters inside the enemy territory, at time t equals 0 with velocity v subscript 0 end subscript equals 250 blank m s to the power of negative 1 end exponent and moves horizontally with constant acceleration a equals 20 m s to the power of negative 2 end exponent (see figure). An enemy tank at the border, spot the plane and fire shots at an angle theta equals 60 degree with the horizontal and with velocity u equals 600 blank m s to the power of negative 1 end exponent. At what altitude H of the plane it can be hit by the shot?

physics-General
If it is being hit then
d equals v subscript 0 end subscript t plus fraction numerator 1 over denominator 2 end fraction a t to the power of 2 end exponent equals left parenthesis u cos invisible function application theta right parenthesis t
or t equals fraction numerator u cos invisible function application theta minus v subscript 0 end subscript over denominator a divided by 2 end fraction

therefore blank t equals fraction numerator 600 cross times fraction numerator 1 over denominator 2 end fraction minus 250 over denominator 10 end fraction equals 5 blank s
H equals open parentheses u sin invisible function application theta close parentheses t minus fraction numerator 1 over denominator 2 end fraction cross times g t to the power of 2 end exponent
equals 600 cross times fraction numerator square root of 3 over denominator 2 end fraction cross times 5 minus fraction numerator 1 over denominator 2 end fraction cross times 10 cross times 25
H equals 2473 blank m
General
physics-

A light ray is incident by grazing one of the face of a prism and after refraction ray does not emerge out, what should be the angle of prism while critical angle is C</em

A light ray is incident by grazing one of the face of a prism and after refraction ray does not emerge out, what should be the angle of prism while critical angle is C</em

physics-General
parallel
General
physics-

A particle of mass m is rotating in a horizontal circle of radius R and is attached to a hanging mass M as shown in the figure. The speed of rotation required by the mass m keep M steady is

To keep the mass M steady, let T is the tension in the string joining the two. Then for particle m comma
T equals fraction numerator m v to the power of 2 end exponent over denominator R end fraction open parentheses i close parentheses
For mass M comma
T equals M g left parenthesis i i right parenthesis
From Eqs. (i) and (ii)
fraction numerator m v to the power of 2 end exponent over denominator R end fraction equals M g ⟹ v equals square root of fraction numerator M g R over denominator m end fraction end root

A particle of mass m is rotating in a horizontal circle of radius R and is attached to a hanging mass M as shown in the figure. The speed of rotation required by the mass m keep M steady is

physics-General
To keep the mass M steady, let T is the tension in the string joining the two. Then for particle m comma
T equals fraction numerator m v to the power of 2 end exponent over denominator R end fraction open parentheses i close parentheses
For mass M comma
T equals M g left parenthesis i i right parenthesis
From Eqs. (i) and (ii)
fraction numerator m v to the power of 2 end exponent over denominator R end fraction equals M g ⟹ v equals square root of fraction numerator M g R over denominator m end fraction end root
General
physics-

Figure shows the variation of the stopping potential (V0) with the frequency (v) of theincident radiations for two different photosensitive material M1 and M2 .What are the values of work functions for M1 and M2 respectively

Figure shows the variation of the stopping potential (V0) with the frequency (v) of theincident radiations for two different photosensitive material M1 and M2 .What are the values of work functions for M1 and M2 respectively

physics-General
General
physics-

The respective angles of the flint and crown glass prisms are A to the power of apostrophe and A. They are to be used for dispersion without deviation, then the ratio of their angles A apostrophe divided by A will be

Since A left parenthesis mu subscript y end subscript minus 1 right parenthesis plus A to the power of ´ end exponent left parenthesis mu subscript y ´ end subscript minus 1 right parenthesis equals 0 rightwards double arrow fraction numerator A to the power of ´ end exponent over denominator A end fraction equals negative open parentheses fraction numerator mu subscript y end subscript minus 1 over denominator mu subscript y ´ end subscript minus 1 end fraction close parentheses

The respective angles of the flint and crown glass prisms are A to the power of apostrophe and A. They are to be used for dispersion without deviation, then the ratio of their angles A apostrophe divided by A will be

physics-General
Since A left parenthesis mu subscript y end subscript minus 1 right parenthesis plus A to the power of ´ end exponent left parenthesis mu subscript y ´ end subscript minus 1 right parenthesis equals 0 rightwards double arrow fraction numerator A to the power of ´ end exponent over denominator A end fraction equals negative open parentheses fraction numerator mu subscript y end subscript minus 1 over denominator mu subscript y ´ end subscript minus 1 end fraction close parentheses
parallel
General
physics-

The two lines A and B shown in figure are the agraphs of the de Broglie wavelength l as function of fraction numerator 1 over denominator square root of V end fraction (V is the accelerating potential) for two particles having the same charge
Which of the two represents the particle of heavier mass ?

The two lines A and B shown in figure are the agraphs of the de Broglie wavelength l as function of fraction numerator 1 over denominator square root of V end fraction (V is the accelerating potential) for two particles having the same charge
Which of the two represents the particle of heavier mass ?

physics-General
General
physics-

In a photoelectric experiment anode potential is plotted against plate current

In a photoelectric experiment anode potential is plotted against plate current

physics-General
General
physics-

From an inclined plane two particles are projected with same speed at same angle theta, one up and other down the plane as shown in figure. Which of the following statements left parenthesis s right parenthesis is/are correct?

Here, alpha equals 2 theta comma beta equals theta

Time of flight of A is, T subscript 1 end subscript equals fraction numerator 2 u sin invisible function application left parenthesis alpha minus beta right parenthesis over denominator g blank c o s blank beta end fraction
equals fraction numerator 2 u sin invisible function application left parenthesis 2 theta minus theta right parenthesis over denominator g blank c o s blank theta end fraction equals fraction numerator 2 u over denominator g end fraction t a n invisible function application theta
Time of flight of B is, T subscript 2 end subscript equals fraction numerator 2 u sin invisible function application theta over denominator g cos invisible function application theta end fraction equals fraction numerator 2 u over denominator g end fraction tan invisible function application theta
So, T subscript 1 end subscript equals T subscript 2 end subscript. The acceleration of both the particles is g downwards. Therefore, relative acceleration between the two is zero or relative motion between the two is uniform. The relative velocity of A w.r.t. B is towards A B blank, therefore collision will take place between the two in mid air.

From an inclined plane two particles are projected with same speed at same angle theta, one up and other down the plane as shown in figure. Which of the following statements left parenthesis s right parenthesis is/are correct?

physics-General
Here, alpha equals 2 theta comma beta equals theta

Time of flight of A is, T subscript 1 end subscript equals fraction numerator 2 u sin invisible function application left parenthesis alpha minus beta right parenthesis over denominator g blank c o s blank beta end fraction
equals fraction numerator 2 u sin invisible function application left parenthesis 2 theta minus theta right parenthesis over denominator g blank c o s blank theta end fraction equals fraction numerator 2 u over denominator g end fraction t a n invisible function application theta
Time of flight of B is, T subscript 2 end subscript equals fraction numerator 2 u sin invisible function application theta over denominator g cos invisible function application theta end fraction equals fraction numerator 2 u over denominator g end fraction tan invisible function application theta
So, T subscript 1 end subscript equals T subscript 2 end subscript. The acceleration of both the particles is g downwards. Therefore, relative acceleration between the two is zero or relative motion between the two is uniform. The relative velocity of A w.r.t. B is towards A B blank, therefore collision will take place between the two in mid air.
parallel
General
Physics-

Two particles 1 and 2 are projected with same speed v as shown in figure. Particle 2 is on the ground and particle 1 is at a height h from the ground and at a horizontal distance s from particle 2. If a graph is plotted between v and s for the condition of collision of the two then (v on y-axis and s on x-axis

Assuming particle 2 to be at rest, substituting in
y equals x tan invisible function application theta minus fraction numerator g x to the power of 2 end exponent over denominator 2 u to the power of 2 end exponent cos to the power of 2 end exponent invisible function application theta end fraction blank left parenthesis theta equals 0 degree right parenthesis

We have negative h equals fraction numerator negative g over denominator 2 left parenthesis 4 v to the power of 2 end exponent right parenthesis end fraction
or v equals square root of fraction numerator g over denominator 8 h end fraction end root
Which is a straight line passing through origin with slopesquare root of fraction numerator g over denominator 8 h end fraction end root

Two particles 1 and 2 are projected with same speed v as shown in figure. Particle 2 is on the ground and particle 1 is at a height h from the ground and at a horizontal distance s from particle 2. If a graph is plotted between v and s for the condition of collision of the two then (v on y-axis and s on x-axis

Physics-General
Assuming particle 2 to be at rest, substituting in
y equals x tan invisible function application theta minus fraction numerator g x to the power of 2 end exponent over denominator 2 u to the power of 2 end exponent cos to the power of 2 end exponent invisible function application theta end fraction blank left parenthesis theta equals 0 degree right parenthesis

We have negative h equals fraction numerator negative g over denominator 2 left parenthesis 4 v to the power of 2 end exponent right parenthesis end fraction
or v equals square root of fraction numerator g over denominator 8 h end fraction end root
Which is a straight line passing through origin with slopesquare root of fraction numerator g over denominator 8 h end fraction end root
General
physics-

a) Name the experiment for which the adjacent graph, showing the variation of intensity of scattered electrons with the angle of scatter ing (q) was obtained.
b) Also name the important hypothesis that was confirmed by this experiment

a) Name the experiment for which the adjacent graph, showing the variation of intensity of scattered electrons with the angle of scatter ing (q) was obtained.
b) Also name the important hypothesis that was confirmed by this experiment

physics-General
General
physics-

Three identical particles are joined together by a thread as shown in figure. All the three particles are moving in a horizontal plane. If the velocity of the outermost particle is v subscript 0 end subscript, then the ratio of tensions in the three sections of the string is

Let omega is the angular speed of revolution

T subscript 3 end subscript equals m omega to the power of 2 end exponent 3 l
T subscript 2 end subscript minus T subscript 3 end subscript equals m omega to the power of 2 end exponent 2 l rightwards double arrow T subscript 2 end subscript equals m omega to the power of 2 end exponent 5 l
T subscript 1 end subscript minus T subscript 2 end subscript equals m omega to the power of 2 end exponent l rightwards double arrow T subscript 1 end subscript equals m omega to the power of 2 end exponent 6 l
T subscript 3 end subscript colon T subscript 2 end subscript colon T subscript 1 end subscript equals 3 blank colon 5 blank colon 6

Three identical particles are joined together by a thread as shown in figure. All the three particles are moving in a horizontal plane. If the velocity of the outermost particle is v subscript 0 end subscript, then the ratio of tensions in the three sections of the string is

physics-General
Let omega is the angular speed of revolution

T subscript 3 end subscript equals m omega to the power of 2 end exponent 3 l
T subscript 2 end subscript minus T subscript 3 end subscript equals m omega to the power of 2 end exponent 2 l rightwards double arrow T subscript 2 end subscript equals m omega to the power of 2 end exponent 5 l
T subscript 1 end subscript minus T subscript 2 end subscript equals m omega to the power of 2 end exponent l rightwards double arrow T subscript 1 end subscript equals m omega to the power of 2 end exponent 6 l
T subscript 3 end subscript colon T subscript 2 end subscript colon T subscript 1 end subscript equals 3 blank colon 5 blank colon 6
parallel
General
physics-

A ball of massopen parentheses m close parentheses 0.5 kg is attached to the end of a string having length left parenthesis L right parenthesis 0.5 m. The ball is rotated on a horizontal circular path about vertical axis. The maximum tension that the string can bear is 324 N. The maximum possible value of angular velocity of ball (in rad/s) is

T cos invisible function application theta component will cancel m g.

T sin invisible function application theta Component will provide necessary centripetal force the ball towards center C.
therefore T sin invisible function application theta equals m r omega to the power of 2 end exponent equals m left parenthesis l sin invisible function application theta right parenthesis omega to the power of 2 end exponent
o r blank T equals m l omega to the power of 2 end exponent ⟹ omega equals square root of fraction numerator T over denominator m l end fraction end root r a d divided by s
o r blank omega subscript m a x end subscript equals square root of fraction numerator T subscript m a x end subscript over denominator m l end fraction equals square root of fraction numerator 324 over denominator 0.5 cross times 0.5 end fraction equals 36 blank r a d divided by s end root end root

A ball of massopen parentheses m close parentheses 0.5 kg is attached to the end of a string having length left parenthesis L right parenthesis 0.5 m. The ball is rotated on a horizontal circular path about vertical axis. The maximum tension that the string can bear is 324 N. The maximum possible value of angular velocity of ball (in rad/s) is

physics-General
T cos invisible function application theta component will cancel m g.

T sin invisible function application theta Component will provide necessary centripetal force the ball towards center C.
therefore T sin invisible function application theta equals m r omega to the power of 2 end exponent equals m left parenthesis l sin invisible function application theta right parenthesis omega to the power of 2 end exponent
o r blank T equals m l omega to the power of 2 end exponent ⟹ omega equals square root of fraction numerator T over denominator m l end fraction end root r a d divided by s
o r blank omega subscript m a x end subscript equals square root of fraction numerator T subscript m a x end subscript over denominator m l end fraction equals square root of fraction numerator 324 over denominator 0.5 cross times 0.5 end fraction equals 36 blank r a d divided by s end root end root
General
maths-

The ellipse x squared over z squared plus y squared over h squared equals 1 and the straight line y = mx + c intersect in real points only if

fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator m to the power of 2 end exponent x to the power of 2 end exponent plus 2 m c x plus c to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction = 1
Þ (b2 + a2 m2) x2 + 2 mca2 x + a2 (c2 - b2) = 0
D > 0 Þ m2c2a4 - a2 (c2 - b2)(b2 + a2m2) > 0 Þ b2 + a2m2 > c2

The ellipse x squared over z squared plus y squared over h squared equals 1 and the straight line y = mx + c intersect in real points only if

maths-General
fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator m to the power of 2 end exponent x to the power of 2 end exponent plus 2 m c x plus c to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction = 1
Þ (b2 + a2 m2) x2 + 2 mca2 x + a2 (c2 - b2) = 0
D > 0 Þ m2c2a4 - a2 (c2 - b2)(b2 + a2m2) > 0 Þ b2 + a2m2 > c2
General
Physics-

A boy throws a cricket ball from the boundary to the wicket-keeper. If the frictional force due to air cannot be ignored, the forces acting on the ball at the position X are respected by

The forces acting on the ball will be (i) in the direction opposite to its motion i e comma frictional force and(ii) weight m g.

A boy throws a cricket ball from the boundary to the wicket-keeper. If the frictional force due to air cannot be ignored, the forces acting on the ball at the position X are respected by

Physics-General
The forces acting on the ball will be (i) in the direction opposite to its motion i e comma frictional force and(ii) weight m g.
parallel

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