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Q.
The angle between the pair of tangents from the point $(1,2)$ to the ellipse $3 x^{2}+2 y^{2}=5$ is
Conic Sections
Solution:
The combined equation of the pair of tangents drawn from $(1,2)$ to the ellipse $3 x^{2}+2 y^{2}=5$ is
$\left(3 x^{2}+2 y^{2}-5\right)(3+8-5)=(3 x+4 y-5)^{2} $
[Using $S S=T^{2}$ ]
$\Rightarrow 9 x^{2}-24 x y-4 y^{2}+\ldots=0$
If angle between these lines is $\theta$,
then $\tan \theta=\frac{2 \sqrt{h^{2}-a b}}{a+b}$,
where $a=9, h=-12, b=-4$
$\Rightarrow \tan \theta=\frac{12}{\sqrt{5}}$
$\Rightarrow \theta=\tan ^{-1}\left(\frac{12}{\sqrt{5}}\right)$