Maths-
General
Easy

Question

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

  1. x squared plus y squared minus 4 y equals 0
  2. x squared plus y squared plus 4 x equals 0
  3. x squared plus y squared minus 4 y equals 12
  4. x squared plus y squared plus 4 x equals 12

Hint:

In this question, we have to find 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. Let the equations be the equation of lines of triangle ABC. Considering AB and BC to find the point B as it is the common point. Similarly, finding point C using AC and BC and finding A using AB and AC. Now we have the co-ordinates of points A, B and C. Further, assuming P (h, k) to be the center of the circle, so from AP = CP and BP = CP we will get two equations using which we can find the co-ordinates of P. Now using a point from A, B or C and P we can find the radius. Further we have the co-ordinates of center and the radius, so we can find the equation of the circle.

The correct answer is: x squared plus y squared minus 4 y equals 12


    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

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