Question

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

• A. $a$
• B. $1 / \sqrt{a}$
• C. $\sqrt{a}$
• D. $1 / a$

Answer 1

Answer: (B)

Length of the perpendicular from the center to the given line,
$\left|\frac{0+0-1}{\sqrt{h^{2}+k^{2}}}\right|=a $
$\Rightarrow \frac{1}{h^{2}+k^{2}}=a^{2}$
$ \Rightarrow h^{2}+k^{2}=\frac{1}{a^{2}}$.
$\therefore$ the locus of $(h, k)$ is
$x^{2}+y^{2}=\frac{1}{a^{2}}$.