Question

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

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

Answer 1

The correct option is B

$3\sqrt{5}$

Calculate the radius of the circle.

Let $m_1\&m_2$ be the slopes of the radius and tangent of the circle.

In the given figure,

$\begin{array}{rcl}{m}_1\times m_2& =& -1\\ \left[\frac{\left\{{\displaystyle \frac{a-4}{2}}\right\}-5}{a-2}\right]\times 2& =& -1\\ \left[\frac{a-14}{2-a}\right]& =& 1\\ 2a& =& 16\\ a& =& 8\end{array}$

Centre of the circle is $\left(8,2\right)$.

Radius of the circle

$=\sqrt{{(8-2)}^2+{(2-5)}^2}$

$$\begin{gathered} =\sqrt{36+9} \\ =\sqrt{45} \\ =3\sqrt{5} \end{gathered}$$

Hence, option $\left(B\right)$ is correct.