Question

If the lengths of the tangents drawn from a point $P$ to the three circles $x^2+y^2-4=0$ $x^2+y^2-2x+3y=0$ and $x^2+y^2+7y-18=0$ 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, $S_1-S_2=0$
$\Rightarrow \left(x^2+y^2-4\right)-\left(x^2+y^2-2x+3y\right)=0$
$\Rightarrow 2x-3y-4=0\dots (i)$
and $S_1-S_3=0$
$\Rightarrow \left(x^2+y^2-4\right)-\left(x^2+y^2+7x-18\right)=0$
$\Rightarrow -7y+14=0$
$\Rightarrow y=2\cdots (ii)$
[From Eqs. (i) and (ii)]
$2x-3(2)-4=0$
$2x-6-4=0$
$\Rightarrow 2x=10$
$\Rightarrow x=5$
So, radical centre $P$ is $(5,2)$.