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The points of intersection of the perpendicular tangents drawn to the ellipse $4 x^2+9 y^2=36$ lie on the curve.
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Verified Answer
The correct answer is:
$x^2+y^2=13$
We have, ellipse
$$
\begin{aligned}
4 x^2+9 y^2 & =36 \\
\Rightarrow \quad \frac{x^2}{9}+\frac{y^2}{4} & =1 \quad \Rightarrow \quad \frac{x^2}{3^2}+\frac{y^2}{2^2}=1
\end{aligned}
$$
So, $\quad a^2=9$ and $b^2=4$.
The points of intersection of the perpendicular tangents lie on director circle.
Equation of director circle is
$$
\begin{array}{ll}
& x^2+y^2=a^2+b^2 \\
\Rightarrow & x^2+y^2=9+4 \\
\Rightarrow & x^2+y^2=13
\end{array}
$$
$$
\begin{aligned}
4 x^2+9 y^2 & =36 \\
\Rightarrow \quad \frac{x^2}{9}+\frac{y^2}{4} & =1 \quad \Rightarrow \quad \frac{x^2}{3^2}+\frac{y^2}{2^2}=1
\end{aligned}
$$
So, $\quad a^2=9$ and $b^2=4$.
The points of intersection of the perpendicular tangents lie on director circle.
Equation of director circle is
$$
\begin{array}{ll}
& x^2+y^2=a^2+b^2 \\
\Rightarrow & x^2+y^2=9+4 \\
\Rightarrow & x^2+y^2=13
\end{array}
$$
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