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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
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Verified Answer
The correct answer is:
(5, 2)
Radical centre is the locus of point $P$ from which equal length of tangent can be drawn to circle.
So, $\quad S_1-S_2=0$
$$
\Rightarrow\left(x^2+y^2-4\right)-\left(x^2+y^2-2 x+3 y\right)=0
$$
and
$$
S_1-S_3=0
$$
$\Rightarrow\left(x^2+y^2-4\right)-\left(x^2+y^2+7 x-18\right)=0$
$\Rightarrow \quad-7 y+14=0$
$$
\begin{gathered}
2 x-3(2)-4=0 \\
2 x-6-4=0 \Rightarrow 2 x=10 \Rightarrow x=5
\end{gathered}
$$
So, radical centre $P$ is $(5,2)$.
So, $\quad S_1-S_2=0$
$$
\Rightarrow\left(x^2+y^2-4\right)-\left(x^2+y^2-2 x+3 y\right)=0
$$
and
$$
S_1-S_3=0
$$
$\Rightarrow\left(x^2+y^2-4\right)-\left(x^2+y^2+7 x-18\right)=0$
$\Rightarrow \quad-7 y+14=0$
$$
\begin{gathered}
2 x-3(2)-4=0 \\
2 x-6-4=0 \Rightarrow 2 x=10 \Rightarrow x=5
\end{gathered}
$$
So, radical centre $P$ is $(5,2)$.
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