First slide

Ellipse

Question

$The angle between the pair of tangents from the point (1, 2) to the ellipse 3 x^{2} + 2 y^{2} = 5 is$

Moderate

A

$\tan^{- 1} (\frac{12}{5})$
B

$\tan^{- 1} (\frac{6}{\sqrt{5}})$
C

$\tan^{- 1} (\frac{12}{\sqrt{5}})$
D

$\tan^{- 1} (12 \sqrt{5})$

Solution

$The combined equation of the pair of tangents drawn from$
$\begin{array}{l} (1, 2) to the ellipse 3 x^{2} + 2 y^{2} = 5 is \\ (3 x^{2} + 2 y^{2} - 5) (3 + 8 - 5) = (3 x + 4 y - 5)^{2} [Using S S^{'} = T^{2}] \\ \Rightarrow 9 x^{2} - 24 x y - 4 y^{2} + \dots = 0 \end{array}$
$\begin{array}{l} If angle between these lines is θ, then \tan θ = \frac{2 \sqrt{h^{2} - a b}}{a + b}, \\ where a = 9, h = - 12, b = - 4 \end{array}$
$\begin{array}{l} \Rightarrow \tan θ = \frac{12}{\sqrt{5}} \\ \Rightarrow θ = \tan^{- 1} (\frac{12}{\sqrt{5}}) \end{array}$

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