Question

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

• A. 45
• B. 75
• C. 20
• D. 50

Answer 1

Answer: (A)

$2 x-y+1=0$ is tangent to the circle
Slope of line $O A=-\frac{1}{2}$
Equation of $O A,(y-5)=-\frac{1}{2}(x-2)$
$2 y-10=-x+2$
$x+2 y=12$
$\therefore$ intersection with $x-2 y=4$ will give coordinates of centre
On solving, we get, the centre of the circle is $(8,2)$
The radius $OA =\sqrt{(8-2)^{2}+(2-5)^{2}}=\sqrt{36+9}$
$=\sqrt{45}=3 \sqrt{5}$ units
Hence, $(O A)^{2}=45$