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JEE Main 2004
Exam
JEE Main
Year
2004
Subject
Mathematics
Chapter
Parabola

JEE Main 2004 · Mathematics · Parabola

A variable circle passes through the fixed point A(p, q) and...

A variable circle passes through the fixed point A(p,q)A(p, q) and touches xx-axis. The locus of the other end of the diameter through A\mathrm{A} is

Options

  1. A(x−p)2=4qy(x-p)^2=4 q y
  2. B(x−q)2=4py(x-q)^2=4 p y
  3. C(y−p)2=4qx(y-p)^2=4 q x
  4. D(y−q)2=4px(y-q)^2=4 p x

Correct answer

A. (x−p)2=4qy(x-p)^2=4 q y

Step-by-step solution

Let the other end of diameter is (h,k)thenequationofcircleis(x−h)(x−p)+(y−k)(y−q)=0Put(\mathrm{h}, \mathrm{k}) then equation of circle is (x-h)(x-p)+(y-k)(y-q)=0 Puty=0,since, sincex−axistouchesthecircle-axis touches the circle⇒x2−(h+p)x+(hp+kq)=0⇒(h+p)2=4(hp+kq)(D=0)⇒(x−p)2=4qy.\begin{aligned}& \Rightarrow x²-(h+p) x+(h p+k q)=0 \Rightarrow(h+p)^2=4(h p+k q) \quad(D=0) \\ & \Rightarrow(x-p)^2=4 q y . \end{aligned}$
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