Question

Find the condition that the straight line $Ax+By+C=0$ may touch the circle.
$(x-a{)}^2+(y-b{)}^2=c^2$.

Answer 1

Fact: If a line is tangent to a circle, then distance of centre of the circle from the line will be same as its radius

Here the center of the circle is $(a,b)$, and radius is $c$

$\Rightarrow \left|\frac{A\left(a\right)+B\left(b\right)+C}{\sqrt{A^2+B^2}}\right|=c$

$\therefore Aa+Bb+C=\pm c\sqrt{A^2+B^2}$