Consider the ellipse Let , , be a point. A straight line drawn through parallel to the -axis crosses the ellipse and its auxiliary circle at points and respectively, in the first quadrant. The tangent to the ellipse at the point intersects the posit

Question

Consider the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1 .$

Let $H (α, 0)$ , $0 < α < 2$ , be a point. A straight line drawn through $H$ parallel to the $y$ -axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$ -axis at a point $G$ . Suppose the straight line joining $F$ and the origin makes an angle $ϕ$ with the positive $x$ -axis. [2022]

	List-I		List-II
(I)	If $ϕ = \frac{π}{4}$ , then the area of the triangle $F G H$ is	(P)	$\frac{{(\sqrt{3} - 1)}^{4}}{8}$
(II)	If $ϕ = \frac{π}{3}$ , then the area of the triangle $F G H$ is	(Q)	$1$
(III)	If $ϕ = \frac{π}{6}$ , then the area of the triangle $F G H$ is	(R)	$\frac{3}{4}$
(IV)	If $ϕ = \frac{π}{12}$ , then the area of the triangle $F G H$ is	(S)	$\frac{1}{2 \sqrt{3}}$
		(T)	$\frac{3 \sqrt{3}}{2}$

The correct option is:

Smart Achievers · Accepted Answer

(3)

$Let F (2 \cos ϕ, 2 \sin ϕ) and E (2 \cos ϕ, \sqrt{3} \sin ϕ)$

$y$

$Tangent at E (2 \cos ϕ, \sqrt{3} \sin ϕ) to ellipse \frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$

$i.e. \frac{x \cos ϕ}{2} + \frac{y \sin ϕ}{\sqrt{3}} = 1$ $intersects x-axis at G (2 \sec ϕ, 0)$

$Area of triangle F G H = \frac{1}{2} H G \times F H$

$= \frac{1}{2} (2 \sec ϕ - 2 \cos ϕ) 2 \sin ϕ;$ $Δ = 2 \sin^{2} ϕ \cdot \tan ϕ$

$Δ = (1 - \cos 2 ϕ) \cdot \tan ϕ$

$I. If ϕ = \frac{π}{4}, Δ = 1 \to (Q)$

$II. If ϕ = \frac{π}{3}, Δ = 2 {(\frac{\sqrt{3}}{2})}^{2} \cdot \sqrt{3} = \frac{3 \sqrt{3}}{2} \to (T)$

$III. If ϕ = \frac{π}{6}, Δ = 2 {(\frac{1}{2})}^{2} \cdot \frac{1}{\sqrt{3}} = \frac{1}{2 \sqrt{3}} \to (S)$

$IV. If ϕ = \frac{π}{12}, Δ = (1 - \frac{\sqrt{3}}{2}) (2 - \sqrt{3}) = \frac{{(2 - \sqrt{3})}^{2}}{2} \to (P)$

Solution

1 (I) → (R); (II) → (S); (III) → (Q); (IV) → (P)

2 (I) → (R); (II) → (T); (III) → (S); (IV) → (P)

3 (I) → (Q); (II) → (T); (III) → (S); (IV) → (P)

4 (I) → (Q); (II) → (S); (III) → (Q); (IV) → (P)

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