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The line $\mathrm{2 x-y+1=0}$ is tangent to the circle at the point $\mathrm{(2,5)}$ and the centre of the circles lies on $\mathrm{x-2 y=4}$ .The radius of the circle is
Option: 1
$3 \sqrt{5}$

Option: 2
$5 \sqrt{3}$

Option: 3
$2 \sqrt{5}$

Option: 4
$5 \sqrt{2}$

Answers (1)

best_answer

$\mathrm{2 \mathrm{x}-\mathrm{y}+1=0}$ is tangent

$\mathrm{\text{slope of line }\mathrm{OA}=-\frac{1}{2}}$
$\mathrm{\text{equation of} \: \mathrm{OA},(\mathrm{y}-5)=-\frac{1}{2}(\mathrm{x}-2)}$
$\mathrm{2 y-10=-x+2}$
$\mathrm{2x+2 y=12}$

$\mathrm{\therefore \quad}$ intersection with $\mathrm{\mathrm{x}-2 \mathrm{y}=4}$ will give coordinates of centre

solving we get $\mathrm{(8,2)}$

distance $\mathrm{\mathrm{OA}=\sqrt{(8-2)^{2}+(2-5)^{2}}=\sqrt{36+9}=\sqrt{45}=3 \sqrt{5}}$

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