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If the line $a x+b y=1$ is a tangent to the circle $s_r \equiv x^2+y^2-r^2=0$, then which one of the following is true?
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The correct answer is:
$(a, b)$ lies on the circle $S_1=0$
We have, $a x+b y=1$ is tangent of circle $x^2+y^2-r^2=0$
$$
\therefore \quad r=\frac{1}{\sqrt{a^2+b^2}} \Rightarrow r^2=\frac{1}{a^2+b^2} \Rightarrow a^2+b^2=\frac{1}{r^2}
$$
$r=1,(a, b)$ lie on the circle
$\therefore(a, b)$ lie on the circle when $S_1=0$
$$
\therefore \quad r=\frac{1}{\sqrt{a^2+b^2}} \Rightarrow r^2=\frac{1}{a^2+b^2} \Rightarrow a^2+b^2=\frac{1}{r^2}
$$
$r=1,(a, b)$ lie on the circle
$\therefore(a, b)$ lie on the circle when $S_1=0$
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