Question

State whether the given statement is True or False. If the line $lx+my=1$ is a tangent to the circle $x^2+y^2=a^2$, then the point $(l,m)$ lies on a circle.

Answer 1

We have circle $x^2+y^2=a^2$, and line $lx+my=1$
Centre of circle is $(0,0)$ and radius is a Since given line is tangent of circle
$\Rightarrow$ Perpendicular distance of line from centre of circle $=$ Radius of circle Perpendicular distance of point
$(x_1,y_1)$ from line $ax+by+c=0$ is
$\frac{|a{x}_1+b{y}_1+c|}{\sqrt{a^2+b^2}}$
$\Rightarrow a=\frac{|l\times 0+m\times 0-1|}{\sqrt{l^2+m^2}}$
$\Rightarrow a=\frac{1}{\sqrt{l^2+m^2}}$
Squaring on both sides
$\Rightarrow l^2+m^2=\frac{1}{a^2}$

Hence, locus of $(l,m)$ is $x^2+y^2=\frac{1}{a^2}$
$\Rightarrow (l,m)$ lies on circle $x^2+y^2=\frac{1}{a^2}$

Hence, the given statement is true.