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, the radius of the circle is:

• A. $4\sqrt{5}$
• B. $5\sqrt{3}$
• C. $3\sqrt{5}$
• D. $5\sqrt{4}$

Answer 1

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

Let the centre be $C=(a,\frac{a-4}{2})$


We know that
$m_1\times m_2=-1\Rightarrow \frac{\frac{a-4}{2}-5}{a-2}\times 2=-1\Rightarrow \frac{a-14}{a-2}=-1\Rightarrow a=8$
Centre $=(8,2)$
Hence, the radius
$=\sqrt{36+9}=\sqrt{45}=3\sqrt{5}$