Question

If the line $hx+ky=\frac{1}{a}$ touches the circle $x^2+y^2=a^2$, then the locus of $(h,k)$ is a circle of radius.

• A. $\frac{1}{a}$
• B. $a^2$
• C. $a$
• D. $\frac{1}{a^2}$

Answer 1

The correct option is D $\frac{1}{a^2}$

Centre of circle $(0,0)$ and radius $=a$

If $hx+ky=\frac{1}{a}$ is tangent to circle, then

distance of centre from line $=$ radius

$\left|\frac{h\left(0\right)+k\left(0\right)-\frac{1}{a}}{\sqrt{h^2+k^2}}\right|=a\left|\frac{-1}{a^2}\right|=\sqrt{h^2+k^2}{h}^2+k^2={\left(\frac{1}{a^2}\right)}^2$

radius $=\frac{1}{a^2}$