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  • The line is tangent to the circle at the point <img alt="(2,5)" src="htt

The line \mathrm{2 x-y+1=0} is tangent to the circle at the point (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

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

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

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



x+2 y=12 \\ -\mathrm{x}-2 \mathrm{y}=4 \\
             +
-----------------------------
\begin{array}{cll} 4 y=8 & \Rightarrow & y=2 \\ x-4=4 & \Rightarrow & x=8 \end{array}

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

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