Ellipse PYQs Set 5 (PDF)

Set 5 of 5 · Questions 121-146 of 146

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Mere Teacher · Previous Year Questions

JEE Main Mathematics · Ellipse · Set 5 (Q121-146)

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121.
If $S$ and $S^{’}$ are the foci of the ellipse $\frac{x ^{2}}{18} + \frac{y ^{2}}{9} = 1$ and P be a point on the ellipse, then $min (SP \cdot S^{’} P) + max (SP \cdot S^{’} P)$ is equal to:
1. (1) $3 (6 + 2)$
2. (2) $3 (1 + 2)$
3. (3) $27$
4. (4) $9$
122.
If $αx + βy = 109$ is the equation of the chord of the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ , whose mid point is
$(\frac{5}{2}, \frac{1}{2})$ , then $α + β$ is equal to:
1. (1) $58$
2. (2) $46$
3. (3) $37$
4. (4) $72$
123.
If m is the slope of a common tangent to the curves $\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1$ and $x^{2}$ + $y^{2}$ = 12, then $12 m^{2}$ is equal to :
1. (1) $6$
2. (2) $9$
3. (3) $10$
4. (4) $12$
124.
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 :
1. (1) $\frac{π}{2} + tan^{- 1} (\frac{1}{4})$
2. (2) $\frac{π}{2} + 2 tan^{- 1} (\frac{1}{4})$
3. (3) $\frac{π}{2} - tan^{- 1} (\frac{1}{4})$
4. (4) $π - tan^{- 1} (\frac{1}{4})$
125.
The tangent and normal to the ellipse $3 x^{2} + 5 y^{2} = 32$ at the point $P (2, 2)$ meet the $x$ -axis at $Q$ and $R$ , respectively. Then the area (in sq. units) of the triangle $PQR$ is :
1. (1) $\frac{34}{15}$
2. (2) $\frac{14}{3}$
3. (3) $\frac{16}{3}$
4. (4) $\frac{68}{15}$
126.
On the ellipse $\frac{x ^{2}}{8} + \frac{y ^{2}}{4} = 1$ let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line x + 2y = 0. Let S and S’ be the foci of the ellipse and be its eccentricity. If A is the area of the triangle SPS' then, the value of (5- $e^{2}$ ). A is :
1. (1) $6$
2. (2) $12 `$
3. (3) $14$
4. (4) $24$
127.
A line passing through the point $P (5, 5)$ intersects the ellipse $\frac{x ^{2}}{36} + \frac{y ^{2}}{25} = 1$ at $A$ and $B$ such that $(PA) \cdot (PB)$ is maximum. Then $5 (P A^{2} + P B^{2})$ is equal to:
1. (1) $290$
2. (2) $377$
3. (3) $338$
4. (4) $218$
128.
The locus of mid-points of the line segments joining (-3,-5)and the points on the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{9} = 1$ is :
1. (1) $9 x^{2} + 4 y^{2} + 18x + 8y + 145 = 0$
2. (2) $36 x^{2} + 16 y^{2} + 90x + 56y + 145 = 0$
3. (3) $36 x^{2} + 16 y^{2} + 108x + 80y + 145 = 0$
4. (4) $36 x^{2} + 16 y^{2} + 72x + 32y + 145 = 0$
129.
Let $P$ be a parabola with vertex $(2, 3)$ and directrix $2 x + y = 6$ . Let an ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$ of eccentricity $\frac{1}{2}$ pass through the focus of the parabola $P$ . Then the square of the length of the latus rectum of $E$ , is
1. (1) $\frac{385}{8}$
2. (2) $\frac{347}{8}$
3. (3) $\frac{512}{25}$
4. (4) $\frac{656}{25}$
130.
Let for two distinct values of $p$ the lines $y = x + p$ touch the ellipse $E : \frac{x ^{2}}{4 ^{2}} + \frac{y ^{2}}{3 ^{2}} = 1$ at the points A and B. Let the line $y = x$ intersect E at the points C and D. Then the area of the quadrilateral $ABCD$ is equal to:
1. (1) $48$
2. (2) $20$
3. (3) $24$
4. (4) $36$
131.
Let $E_{1} : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$ . Let $E_{2}$ be another ellipse such that it touches the end points of major axis of $E_{1}$ and the foci of $E_{2}$ are the end points of minor axis of $E_{1}$ . If $E_{1}$ and $E_{2}$ have same eccentricities, then its value is :
1. (1) $\frac{- 1 + 5}{2}$
2. (2) $\frac{- 1 + 8}{2}$
3. (3) $\frac{- 1 + 3}{2}$
4. (4) $\frac{- 1 + 6}{2}$
132.
Consider ellipses $E_{k}$ : $k x^{2}$ + $k^{2} y^{2}$ = 1, k = 1, 2,..., 20. Let $C_{k}$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $E_{k}$ . If $r_{k}$ is the radius of the circle $C_{k}$ , then the value of i
1. (1) $3080$
2. (2) $2870$
3. (3) $3210$
4. (4) $3320$
133.
The locus of the foot of perpendicular drawn from the centre of the ellipse x2 + 3y2 = 6 on any tangent to it is:
1. (1) $(x2 + y2)2 = 6x2 + 2y2$
2. (2) $(x2 + y2)2 = 6x2 - 2y2$
3. (3) $(x2 - y2)2 = 6x2 + 2y2$
4. (4) $(x2 - y2)2 = 6x2 - 2y2$
134.
$For the ellipse \frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1, the lines given in List-I are to be matched with their equations given in List-II. : :$
$List-I$ $List-II$
$(P) Directrix corresponding to the focus (- 3, 0);$ $(1) y = 4;$
$(Q) Tangent at the vertex (0, 4);$ $(2) 3 x = 25;$
$(R) Latus rectum through (3, 0) .$ $(3) x = 3;$
$(4) y + 4 = 0;$
$(5) x + 3 = 0;$
$(6) 3 x + 25 = 0 .$
$The correct answer is$
1. (1) $P-2, Q-1, R-5$
2. (2) $P-6, Q-1, R-3$
135.
If distance between the foci of an ellipse is 6 and distance between its directrices is 12, then length of its latus rectum is
1. (1) $4$
2. (2) $32$
3. (3) $9$
4. (4) $22$
136.
Let the ellipse $E_{1} : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$ and $E_{2} : \frac{x ^{2}}{A ^{2}} + \frac{y ^{2}}{B ^{2}} = 1, A < B$ have same eccentricity $\frac{1}{3}$ .
Let the product of their lengths of latus rectums be $\frac{32}{3}$ , and the distance between the foci of $E_{1}$ be 4. If $E_{1}$ and $E_{2}$ meet at $A, B, C$ and $D$ , then the area of the quadrilateral $ABCD$ equals:
1. (1) $\frac{12 6}{5}$
2. (2) $66$
3. (3) $\frac{18 6}{5}$
4. (4) $\frac{24 6}{5}$
137.
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 $2 l_{1}^{2} = 9 l_{2}$ . If the distance between the foci of $E_{1}$ is 8 , then the distance between the foci of $E_{2}$ is
1. (1) $\frac{32}{5}$
2. (2) $\frac{8}{5}$
3. (3) $\frac{16}{5}$
4. (4) $\frac{96}{5}$
138.
Let $S \equiv \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} - 1 = 0, S^{’} \equiv \frac{x ^{2}}{α ^{2}} + \frac{y ^{2}}{β ^{2}} - 1 = 0$ be two intersecting ellipses. If $p (a cos θ, b sin θ)$ and $Q (a cos (\frac{π}{2} + θ), b sin (\frac{π}{2} + θ))$ are their points of intersection then $\frac{1}{2} (a^{2} β^{2} + b^{2} α^{2}) =$
1. (1) $a^{2} b^{2}$
2. (2) $α^{2} + β^{2}$
3. (3) $a^{2} + b^{2}$
4. (4) $α^{2} β^{2}$
139.
Let $A (α, 0)$ and $B (0, β)$ be the points on the line $5 x + 7 y = 50$ . Let the point $P$ divide the line segment $AB$ internally in the ratio $7 : 3$ . Let $3 x - 25 = 0$ be a directrix of the ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ and the corresponding focus be $S$ . If from $S$ , the perpendicular on the $x$ -axis passes through $P$ , then the length of the latus rectum of $E$ is equal to
1. (1) $\frac{25}{3}$
2. (2) $\frac{32}{9}$
3. (3) $\frac{25}{9}$
4. (4) $\frac{32}{5}$
140.
The center of a circle C is at the center of the ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$ . Let C pass through the foci $F_{1}$ and $F_{2}$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let P be one of these four points. If the area of the triangle $PF_{1} F_{2}$ is 30 and the length of the major axis of $E$ is 17, then the distance between the foci of $E$ is:
1. (1) $12$
2. (2) $26$
3. (3) $13$
4. (4) $\frac{13}{2}$
141.
Let $C$ be the circle of minimum area enclosing the ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ with eccentricity $\frac{1}{2}$ and foci $(\pm 2, 0)$ . Let $PQR$ be a variable triangle, whose vertex $P$ is on the circle $C$ and the side $QR$ of length $2 a$ is parallel to the major axis of $E$ and contains the point of intersection of $E$ with the negative $y$ -axis. Then the maximum area of the triangle $PQR$ is:
1. (1) $8 (3 + 2)$
2. (2) $8 (2 + 3)$
3. (3) $6 (3 + 2)$
4. (4) $6 (2 + 3)$
142.
The locus of the point of intersection of perpendicular tangents to the ellipse is called
1. (1) $director circle$
2. (2) $hyperbola$
3. (3) $ellipse$
4. (4) $auxiliary circle$
143.
$Let C be the largest circle centred at (2, 0) and inscribed in the ellipse = \frac{x ^{2}}{36} + \frac{y ^{2}}{16} = 1 . If (1, α) lies on C, then 10 α^{2} is equal to$
144.
$Let the tangents at the points P and Q on the ellipse \frac{x ^{2}}{2} + \frac{y ^{2}}{4} = 1 meet at the point R (2, 22 - 2) . If S is the focus of the ellipse on its negative major axis, then S P^{2} + S Q^{2} is equal to$
145.
$Let S =_{(x, y_{)} \in N \times N : 9_{(} x - 3_{)}^{2} + 1 6_{(} y - 4_{)}^{2} \leq 144} And T =_{(x, y_{)} \in R \times R :_{(} x - 7_{)}^{2} +_{(} y - 4_{)}^{2} \leq 36} . The n_{(} S \cap T_{)} is equal to____.$
146.
$Let an ellipse with centre (1, 0) and latus rectum of length \frac{1}{2} have its major axis along x-axis. If its minor axis subtends an angle 60° at the foci, then the square of the sum of the lengths of its minor and major axes is equal to_______.$