Question

The line $2x-y+1=0$ is tangent to a circle at point $(2,5)$ and the centre of the circle lies on $x-2y=4.$ Then the radius of the circle is ____ units.

• A. $3\sqrt{5}$
• B. $5\sqrt{3}$
• C. $2\sqrt{5}$
• D. $5\sqrt{2}$

Answer 1

The correct option is A $3\sqrt{5}$


Let $O$ be the centre of circle $S.$
As $2x-y+1=0$ is tangent to $S,$
$\therefore m_{OA}\times 2=-1\Rightarrow m_{OA}=-\frac{1}{2}$

Equation of $OA$ is
$(y-5)=-\frac{1}{2}(x-2)$
$\Rightarrow x+2y=12\dots \left(1\right)$

Now, intersection of equation $\left(1\right)$ with $x-2y=4$ will give coordinates of centre $\left(O\right)$ which are $(8,2)$
$\therefore r=OA=\sqrt{(8-2{)}^2+(2-5{)}^2}$
$\Rightarrow r=3\sqrt{5}$ units