First slide

Ellipse

Question

$If x \cos α + y \sin α = 4 is tangent to \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1, then the value of α is$

Moderate

A

$\tan^{- 1} (3 / \sqrt{7})$
B

$\tan^{- 1} (7 / 3)$
C

$\tan^{- 1} (\sqrt{3} / 7)$
D

$\tan^{- 1} (3 / 7)$

Solution

$x \cos α + y \sin α = 4$
$\begin{array}{l} ∴ y = (- \cot α) x + 4 cosec α \\ ∴ m = - \cot α, c = 4 cosec α, a^{2} = 25, b^{2} = 9 \end{array}$
$Now c^{2} = a^{2} m^{2} + b^{2}$
$\begin{array}{l} ∴ 16 {cosec}^{2} α = 25 \cot^{2} α + 9 \\ ∴ 16 (1 + \cot^{2} α) = 25 \cot^{2} α + 9 \\ ∴ 7 = 9 \cot^{2} α \Rightarrow \cot α = \frac{\sqrt{7}}{3} \Rightarrow \tan α = \frac{3}{\sqrt{7}} \\ ∴ α = \tan^{- 1} (3 / \sqrt{7}) \end{array}$

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