Question

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

• A. ${(y-q)}^2=4px$
• B. ${(x-q)}^2=4py$
• C. ${(y-p)}^2=4qx$
• D. ${(x-p)}^2=4qy$

Answer 1

The correct option is D ${(x-p)}^2=4qy$

Let the variable circle be

$x^2+y^2+2gx+2fy+c=0$ ...(1)

$\therefore p^2+q^2+2gp+2fy+c=0$ ...(2)

Circle (1) touches x-axis,

$\therefore g^2-c=0\Rightarrow c=g^2$

From (2) $p^2+q^2+2gp+2fq+g^2=0$ ...(3)

Let the other end of diameter through $(p,q)$ be $(h,k)$, then

$\frac{h+p}{2}=-g$ and $\frac{k+q}{2}=-f$

Put in (3)

$p^2+q^2+2p(-\frac{h+p}{2})+2q(-\frac{k+q}{2})+{\left(\frac{h+p}{2}\right)}^2=0\Rightarrow h^2+p^2-2hp-4kq=0$

$\therefore$ Locus of $(h,k)$ is

$x^2+p^2-2xp-4yk=0\Rightarrow {(x-p)}^2=4qy$