First slide

Ellipse

Question

If the tangent at $P (θ)$ on the ellipse $16 x^{2} + 11 y^{2} = 256$
touches the circle $x^{2} + y^{2} + 2 x - 15 = 0,$ then $θ -$

Moderate

A

$\frac{π}{6}$
B

$\frac{π}{3}$
C

$\frac{2 π}{3}$
D

$\frac{5 π}{6}$

Solution

Centre of the circle is (– 1, 0) and radius is 4. Equation of the tangent at $P (θ)$ to the ellipse is
$\frac{x}{4} \cos θ + \frac{y}{16 / \sqrt{11}} \sin θ = 1$
If this touches the circle, then
$\begin{array}{l} \frac{\frac{- \cos θ}{4} - 1}{\sqrt{\frac{\cos^{2} θ}{16} + \frac{\sin^{2} θ}{256 / 11}}} = 4 \\ \Rightarrow (\cos θ + 4)^{2} = 256 (\frac{\cos^{2} θ}{16} + \frac{11 \sin^{2} θ}{256}) \\ \Rightarrow 4 \cos^{2} θ - 8 \cos θ - 5 = 0 \\ \Rightarrow (2 \cos θ - 5) (2 \cos θ + 1) = 0 \\ \Rightarrow \cos θ = - 1 / 2 \Rightarrow θ = 2 π / 3 \end{array}$

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