Q77 / 170 · Manipal (MET) 2016 · Mathematics

The equation of the circle passing through the point $(2 a, 0)$ and whose radical axis is $x=\frac{a}{2}$ with respect to the circle $x^2+y^2=a^2$, will be

  • Correct answer
    A
    $x^2+y^2-2 a x=0$
  • B
    $x^2+y^2+2 a x=0$
  • C
    $x^2+y^2+2 a y=0$
  • D
    $x^2+y^2-2 a y=0$

Solution

Let the equation of circle be $x^2+y^2+2 g x+2 f y+c=0 .$ Since, it passes through the point $(2 a, 0)$, so $4 a^2+4 a g+c=0$\ldots(i) Also, its radical axis with $x^2+y^2=a^2$ is $2 g x+2 f y+c+a^2=0$ But the radical axis is $x=\frac{a}{2}$, so we get $2 g x+2 f y+c+a^2=0$ But the radical axis is $x=\frac{a}{2}$, so we get $\frac{2 g}{1}=\frac{c+a}{-a / 2}$ and $f=0$ or $a g+c+a=0$\ldots(ii) From Eqs. (i) and (ii), we get $g$ and $c$ Hence, the equation is $x^2+y^2-2 a x=0$

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

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