Q88 / 160 · TS EAMCET 2024 (11 May Shift 1) · Mathematics

A circle passes through the points $(2,0)$ and $(1,2)$. If the power of the point $(0,2)$ with respect to this circle is 4 , then the radius of the circle is

  • A
    $2$
  • Correct answer
    B
    $\sqrt{\frac{5}{2}}$
  • C
    $\sqrt{5}$
  • D
    $4$

Solution

Let the equation of circle be $x^2+y^2+2 g x+2 f y+c=0$ $(2,0)$ and $(1,2)$ lies on it. $\Rightarrow 4+4 g+c=0 \Rightarrow 4 g+c=-4$ ...(i) $1+4+2 g+4 f+c=0 \Rightarrow 2 g+4 f+c=-5$ ...(ii) $4=S_1^2=4+4 f+c \Rightarrow 4 f+c=0$ ...(iii) Solving (i), (ii) and (iii) we get $\begin{aligned} & g=\frac{-5}{2}, c=6, f=\frac{-6}{4}=\frac{-3}{2} \\ & \text { Radius }=\sqrt{g^2+f^2-c}=\sqrt{\frac{25}{4}+\frac{9}{4}-6}=\sqrt{\frac{17}{2}-6}=\sqrt{\frac{5}{2}} . \end{aligned}$

Hence, the correct answer is option: 'B'.

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