JEE Exam  >  JEE Questions  >  A variable circle passes through the fixed po... Start Learning for Free
A variable circle passes through the fixed point A( p,q) and touches x-axis . The locus of the other end of the diameter through A is [2004]
  • a)
    ( y - q)=4 px
  • b)
    ( x - q)=4 py
  • c)
    ( y - p)=4qx
  • d)
    ( x - p)=4qy
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
A variable circle passes through the fixed point A( p,q) and touches x...
Let the variable circle be x2 + y2 + 2gx + 2fy + c = 0 ....(1)
∴ p2 + q2 + 2gp + 2fq + c =0 ....(2)
Circle (1) touches x-axis,
∴ g2 - c = 0 ⇒ c=g2 .
From (2) p2 + q2 + 2gp + 2fq +g=0 ....(3)
Let the other end of diameter through (p, q) be (h, k),
then 
Put in (3)
⇒ h2 + p2 - 2hp - 4kq=0
∴ locus of (h, k) is x2 + p2 - 2 xp - 4 yq=0
⇒ ( x - p )2=4qy
View all questions of this test
Most Upvoted Answer
A variable circle passes through the fixed point A( p,q) and touches x...
Problem Analysis:
We are given a variable circle that passes through a fixed point A(p, q) and touches the x-axis. We need to determine the locus of the other end of the diameter through A.

Solution:
Let B(x, y) be the other end of the diameter through A.

Step 1: Find the equation of the circle:
Since the circle passes through A(p, q), the equation of the circle is given by:
(x - p)^2 + (y - q)^2 = r^2

where r is the radius of the circle.

Step 2: Find the equation of the tangent:
Since the circle touches the x-axis, the tangent at the point of contact will be parallel to the y-axis. Therefore, the equation of the tangent is given by:
x = p

Step 3: Find the coordinates of the point of contact:
Substituting x = p in the equation of the circle, we get:
(p - p)^2 + (y - q)^2 = r^2
0 + (y - q)^2 = r^2
(y - q)^2 = r^2

Therefore, the point of contact is (p, q).

Step 4: Find the coordinates of the other end of the diameter:
Since the diameter passes through the center of the circle, the coordinates of the center are (p, q). Therefore, the other end of the diameter is the reflection of A(p, q) with respect to the center (p, q). This can be obtained by using the formula for reflection:

x' = 2p - x
y' = 2q - y

Therefore, the coordinates of B(x, y) are:
x = 2p - p = p
y = 2q - q = q

Step 5: Find the equation of the locus:
Substituting x = p and y = q in the equation of the circle, we get:
(x - p)^2 + (y - q)^2 = r^2
(p - p)^2 + (q - q)^2 = r^2
0 + 0 = r^2
0 = r^2

Therefore, the equation simplifies to:
(x - p)^2 + (y - q)^2 = 0

Simplifying further, we get:
(x - p)^2 = 0

Therefore, the locus of the other end of the diameter through A is:
(x - p)^2 = 0

which can be written as:
(x - p)^2 = 4qy

Hence, the correct answer is option D: (x - p)^2 = 4qy.
Explore Courses for JEE exam
Question Description
A variable circle passes through the fixed point A( p,q) and touches x-axis . The locus of the other end of the diameter through A is [2004]a)( y - q)2=4 pxb)( x - q)2=4 pyc)( y - p)2=4qxd)( x - p)2=4qyCorrect answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A variable circle passes through the fixed point A( p,q) and touches x-axis . The locus of the other end of the diameter through A is [2004]a)( y - q)2=4 pxb)( x - q)2=4 pyc)( y - p)2=4qxd)( x - p)2=4qyCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A variable circle passes through the fixed point A( p,q) and touches x-axis . The locus of the other end of the diameter through A is [2004]a)( y - q)2=4 pxb)( x - q)2=4 pyc)( y - p)2=4qxd)( x - p)2=4qyCorrect answer is option 'D'. Can you explain this answer?.
Solutions for A variable circle passes through the fixed point A( p,q) and touches x-axis . The locus of the other end of the diameter through A is [2004]a)( y - q)2=4 pxb)( x - q)2=4 pyc)( y - p)2=4qxd)( x - p)2=4qyCorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE. Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of A variable circle passes through the fixed point A( p,q) and touches x-axis . The locus of the other end of the diameter through A is [2004]a)( y - q)2=4 pxb)( x - q)2=4 pyc)( y - p)2=4qxd)( x - p)2=4qyCorrect answer is option 'D'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of A variable circle passes through the fixed point A( p,q) and touches x-axis . The locus of the other end of the diameter through A is [2004]a)( y - q)2=4 pxb)( x - q)2=4 pyc)( y - p)2=4qxd)( x - p)2=4qyCorrect answer is option 'D'. Can you explain this answer?, a detailed solution for A variable circle passes through the fixed point A( p,q) and touches x-axis . The locus of the other end of the diameter through A is [2004]a)( y - q)2=4 pxb)( x - q)2=4 pyc)( y - p)2=4qxd)( x - p)2=4qyCorrect answer is option 'D'. Can you explain this answer? has been provided alongside types of A variable circle passes through the fixed point A( p,q) and touches x-axis . The locus of the other end of the diameter through A is [2004]a)( y - q)2=4 pxb)( x - q)2=4 pyc)( y - p)2=4qxd)( x - p)2=4qyCorrect answer is option 'D'. Can you explain this answer? theory, EduRev gives you an ample number of questions to practice A variable circle passes through the fixed point A( p,q) and touches x-axis . The locus of the other end of the diameter through A is [2004]a)( y - q)2=4 pxb)( x - q)2=4 pyc)( y - p)2=4qxd)( x - p)2=4qyCorrect answer is option 'D'. Can you explain this answer? tests, examples and also practice JEE tests.
Explore Courses for JEE exam

Top Courses for JEE

Explore Courses
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev