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The line $l x+m y+n=0$ will be a tangent to the circle $x^2+y^2=a^2$ iff
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Verified Answer
The correct answer is:
$a^2\left(l^2+m^2\right)=n^2$
Line $y=m x+c$ is tangent, if $c= \pm a \sqrt{1+m^2}$.
Now $1 x+m y+n=0$ or $y=-\frac{1}{m} x-\frac{n}{m}$ is tangent, if
$-\frac{n}{m}= \pm a \sqrt{1+\left(\frac{l}{m}\right)^2} \text { or } n^2=a^2\left(m^2+l^2\right) \text {. }$
Now $1 x+m y+n=0$ or $y=-\frac{1}{m} x-\frac{n}{m}$ is tangent, if
$-\frac{n}{m}= \pm a \sqrt{1+\left(\frac{l}{m}\right)^2} \text { or } n^2=a^2\left(m^2+l^2\right) \text {. }$
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