Question

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

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

Answer 1

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

The equation of the circle with $2x-y+1=0$ as tangent at the point

$(2,5)$ is $S+\lambda P=0$ where $S$ is point circle at point of contact and $P$ is the tangent.
i.e ${(x-2)}^2+{(y-5)}^2+\lambda (2x-y+1)=0$

$\Rightarrow x^2+y^2+(2\lambda -4)x-(\lambda +10)y+29+\lambda =0$
The given centre of the circle lies on $x-2y=4$

$\Rightarrow (-\lambda +2,\frac{\lambda +10}{2})$ lies on $x-2y=4$

$\Rightarrow \lambda =-6$

$\Rightarrow$ equation of circle is $x^2+y^2-16x-4y+23=0$
$\therefore r=\sqrt{g^2+f^2-c}=3\sqrt{5}$