Question

$A$ point $P$ is taken on the circle $x^2+y^2=a^2$.$PN,PM$ are drawn perpendicular to axes. The locus of the pole of line $MN$ is

• A. $x^2+y^2=a^2$
• B. $x^{-2}+y^{-2}=a^{-2}$
• C. $x^{-2}-y^{-2}=a^{-2}$
• D. $x^{-2}+y^{-2}=a^{-4}$

Answer 1

Answer: (B)

$x^{-2}+y^{-2}=a^{-2}$

$M\equiv (0,asin\theta )andN\equiv (acos\theta ,0)$
$\therefore$ Let $A(h,k)$ is pole of line $MN$.
Then, chord of contact from $A(h,k)$ to circle $x^2+y^2=a^2$ is
$hx+ky=a^2$ ..... (1)
And equation of MN is
$y=\frac{-sin\theta }{cos\theta }(x-acos\theta )$
$ycos\theta +sin\theta x=acos\theta sin\theta$ ...... (2)
Equations (1) and (2) represent the same line, then
$\frac{k}{cos\theta }=\frac{h}{sin\theta }=\frac{a^2}{asin\theta cos\theta }$
$\therefore k=acosec\theta$ and $h=asec\theta$
$\therefore$ Equation of locus of $A(h,k)$ is
$\frac{a^2}{h^2}+\frac{a^2}{k^2}=1$
Replace $h\to x$ and $k\to y$
$\Rightarrow x^{-2}+y^{-2}=a^{-2}$