Question

lf the pole of a line w.r. $t$. the circle $x^2+y^2=a^2$ lies on the circle $x^2+y^2=a^4$ , then the line touches the circle

• A. $x^2+y^2=2$
• B. $x^2+y^2=1$
• C. $x^2+y^2=3$
• D. $x^2+y^2=4$

Answer 1

Answer: (B)

$x^2+y^2=1$

Let any point on circle $x^2+y^2=a^4$ be $P(a^2\cos {\theta }_1,a^2\sin \theta )$
Then equation of chord of contact from P on circle $x^2+y^2=a^2$ is
$x{a}^2\cos \theta +y{a}^2\sin \theta =a^2$
$x\cos \theta +y\sin \theta =1-----\left(1\right)$
$x\cos \theta +y\sin \theta =1$ is an equation of polar of $x^2+y^2=0$
So,$x\cos \theta +y\sin \theta =1$ is tangent to a circle $x^2+y^2=1$