Question

If the lengths of the tangents drawn from a point $P$ to the three circles $x^{2}+y^{2}-4=0$ $x^{2}+y^{2}-2 x+3 y=0$ and $x^{2}+y^{2}+7 y-18=0$ are equal, then the coordinates of $P$ are

• A. (2,5)
• B. (3,4)
• C. (4,3)
• D. (5,2)

Answer 1

Answer: (D)

Radical centre is the locus of point $P$ from which equal length of tangent can be drawn to circle.
So, $S_{1}-S_{2}=0$
$\Rightarrow \left(x^{2}+y^{2}-4\right)-\left(x^{2}+y^{2}-2 x+3 y\right)=0$
$\Rightarrow 2 x-3 y-4=0 \,\,\,\ldots(i)$
and $S_{1}-S_{3}=0$
$\Rightarrow \left(x^{2}+y^{2}-4\right)-\left(x^{2}+y^{2}+7 x-18\right)=0$
$\Rightarrow -7 y+14=0$
$\Rightarrow y=2 \,\,\,\cdots(ii)$
[From Eqs. (i) and (ii)]
$2 x-3(2)-4=0$
$2 x-6-4=0$
$ \Rightarrow 2 x=10$
$ \Rightarrow x=5$
So, radical centre $P$ is $(5,2)$.