1. Tardigrade
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  4. If the line h x+k y=1 touches the circle x2+y2=a2 then the locus of the point (h, k) is a circle of radius

Q. 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

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Solution:

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
$x^{2}+y^{2}=\frac{1}{a^{2}}$.

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