Question

Find the condition that the straight line $\frac{1}{r}=a\cos \theta +b\sin \theta$ may touch the circle $r=2c\cos \theta$.

Answer 1

For the line,

$x=r\cos \theta$

$y=r\sin \theta$

So, it becomes

$\frac{1}{r}=\frac{ax}{r}+\frac{by}{r}$

$\Rightarrow ax+by=1$

For the circle

$x=r\cos \theta$

$y=r\sin \theta$

$r=\frac{2x}{r}$

$\Rightarrow x^2+y^2=2x$

$\Rightarrow (x-1{)}^2+y^2=1$

Centre $(1,0)$ and radius $=1$

For $ax+by-1=0$ to be tangent, perpendicular distance from $(1,0)$ must be equal to $1$

$\left|\frac{a-1}{\sqrt{a^2+b^2}}\right|=1$

$\Rightarrow (a-1{)}^2=a^2+b^2$

$\Rightarrow b^2+2a=1$