The coordinates of A and B are, Tangents are drawn from the point P(3, 4) to the ellipse touching the ellipse at points A and B. [2010]

Question

Tangents are drawn from the point P(3, 4) to the ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ touching the ellipse at points A and B. [2010]

Q. The coordinates of A and B are

Smart Achievers · Accepted Answer

(4)

$Tangent to \frac{x^{2}}{3^{2}} + \frac{y^{2}}{2^{2}} = 1 at the point (3 \cos θ, 2 \sin θ) is$

$\frac{x \cos θ}{3} + \frac{y \sin θ}{2} = 1$

$Since it passes through (3, 4),$

$∴$ $\cos θ + 2 \sin θ = 1$

$\Rightarrow 4 \sin^{2} θ = 1 + \cos^{2} θ - 2 \cos θ$

$\Rightarrow 5 \cos^{2} θ - 2 \cos θ - 3 = 0$

$\Rightarrow \cos θ = 1, - \frac{3}{5}$ $\Rightarrow \sin θ = 0, \frac{4}{5}$

$∴$ $Required points are A (3, 0) and B (- \frac{9}{5}, \frac{8}{5})$

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