Ellipse-JEE-Mains-PYQ’s

An ellipse has its center at $(1, -2)$ , one focus at $(3, -2)$ and one vertex at $(5, -2)$ . Then the length of its latus rectum is :

(A) $6$

(B) $6\sqrt{3}$

(D) $4\sqrt{3}$

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Let the length of the latus rectum of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, (a > b)$ , be $30$ . If its eccentricity is the maximum value of the function $f(t) = -\frac{3}{4} + 2t - t^2$ , then $(a^2 + b^2)$ is equal to

(A) $276$

(B) $516$

(D) $496$

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Let each of the two ellipses $E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, (a > b)$ and $E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, (A < B)$ have eccentricity $\frac{4}{5}$ . Let the lengths of the latus recta of $E_1$ and $E_2$ be $l_1$ and $l_2$ , respectively, such that $2l_1^2 = 9l_2$ . If the distance between the foci of $E_1$ is $8$ , then the distance between the foci of $E_2$ is

(A) $\frac{96}{5}$

(B) $\frac{8}{5}$

(D) $\frac{32}{5}$

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If the points of intersection of the ellipses $x^2 + 2y^2 - 6x - 12y + 23 = 0$ and $4x^2 + 2y^2 - 20x - 12y + 35 = 0$ lie on a circle of radius $r$ and centre $(a, b)$ , then the value of $ab + 18r^2$ is :

(A) $53$

(B) $52$

(D) $51$

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Let the line $y - x = 1$ intersect the ellipse $\frac{x^2}{2} + \frac{y^2}{1} = 1$ at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is :

(A) $\pi - \tan^{-1}\left(\frac{1}{4}\right)$

(B) $\frac{\pi}{2} + \tan^{-1}\left(\frac{1}{4}\right)$

(D) $\frac{\pi}{2} - \tan^{-1}\left(\frac{1}{4}\right)$

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Let S and S $'$ be the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ and P $(\alpha, \beta)$ be a point on the ellipse in the first quadrant. If $(\text{SP})^2 + (\text{S}'\text{P})^2 - \text{SP} \cdot \text{S}'\text{P} = 37$ , then $\alpha^2 + \beta^2$ is equal to :

(A) $13$

(B) $15$

(D) $17$

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If the line $\alpha x + 4y = \sqrt{7}$ , where $\alpha \in \mathbf{R}$ , touches the ellipse $3x^2 + 4y^2 = 1$ at the point P in the first quadrant, then one of the focal distances of $P$ is :

(A) $\frac{1}{\sqrt{3}} - \frac{1}{2\sqrt{11}}$

(B) $\frac{1}{\sqrt{3}} - \frac{1}{2\sqrt{5}}$

(D) $\frac{1}{\sqrt{3}} + \frac{1}{2\sqrt{7}}$

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Let $O$ be the vertex of the parabola $x^{2}=4y$ and $Q$ be any point on it. Let the locus of the point $P$ , which divides the line segment $OQ$ internally in the ratio $2:3$ be the conic $C$ . Then the equation of the chord of $C$ , which is bisected at the point $(1, 2)$ , is :

(A) $5x-4y+3=0$

(B) $x-2y+3=0$

(D) $5x-y-3=0$

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An ellipse has its center at $(1, -2)$ , one focus at $(3, -2)$ and one vertex at $(5, -2)$ . Then the length of its latus rectum is :

(A) 6

(B) $6\sqrt{3}$

(D) $4\sqrt{3}$

Click to View Answer

Correct Answer: [ ]

Let the length of the latus rectum of an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, (a > b)$ , be 30. If its eccentricity is the maximum value of the function $f(t) = -\frac{3}{4} + 2t - t^{2}$ , then $(a^{2} + b^{2})$ is equal to

(A) 276

(B) 516

(D) 496

Click to View Answer

Correct Answer: [ ]

Ellipse-JEE-Mains-PYQ’s

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