Question

The tangents PA and PB are drawn from any point P of the circle $x^2+y^2=2a^2$ to the circle $x^2+y^2=a^2.$ The chord of contact AB on extending meets again the first circle at the point A’ and B’. The locus of point of intersection of tangents at A’ and B’ may be given as

• A. $x^2+y^2=8a^2$
• B. $x^2+y^2=12a^2$
• C. $x^2+y^2=4a^2$
• D. $x^2+y^2=16a^2$

Answer 1

The correct option is A $x^2+y^2=8a^2$


Equation of chord of contacts of tangents drawn from any point $P(\sqrt{2}acos\theta ,\sqrt{2}asin\theta )$ to circle $x^2+y^2=a^2$ is
$xcos\theta +ysin\theta =\frac{a}{\sqrt{2}}$ ....(1)
Let Q(h, k) be the point of intersection of tangents from A’ and B’
$\therefore$ Eqn. A'B' is $hx+ky=2a^2$ ...(2)
(1) and (2) represents same line
$\therefore \frac{cos\theta }{h}=\frac{sin\theta }{k}=\frac{1}{2\sqrt{2}a}\Rightarrow cos\theta =\frac{h}{2\sqrt{2}a},sin\theta =\frac{k}{2\sqrt{2}a}$
$\therefore \frac{h^2}{8a^2}+\frac{k^2}{8a^2}=1\therefore$ locus of (h, k) is $x^2+y^2=8a^2$