Ellipse PYQs Set 4 (PDF)

Set 4 of 5 · Questions 91-120 of 146

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Ellipse · Set 4 (Q91-120)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
91. $If an ellipse with its axes as coordinate axes, 2 a and 2 b as the lengths of its major and minor axes respectively passes through the points (2, 2) and (3, 1), then 3 a^{2} + 5 b^{2} =$ (1) $32$ (2) $8$ (3) $64$ (4) $16$ 92. $Let the ellipse 3 x^{2} + p y^{2} = 4 pass through the centre C of the circle x^{2} + y^{2} - 2 x - 4 y - 11 = 0 of radius r . Let f_{1}, f_{2} be the focal distances of the point C on the ellipse. Then 6 f_{1} f_{2} - r is equal to$ (1) $78$ (2) $68$ (3) $70$ (4) $74$ 93. $An ellipse has its center at (1, - 2), one focus at (3, - 2) and one vertex at (5, - 2) . Then the length of its latus rectum is:$ (1) $6$ (2) $43$ (3) $\frac{16}{3}$ (4) $63$ 94. $An ellipse, with foci at (0, 2) and (0, - 2) and minor axis of length 4, passes through which of the following points?$ (1) $(2, 2)$ (2) $(2, 2)$ (3) $(2, 22)$ (4) $(1, 22)$ 95. $For some θ \in (0, \frac{π}{2}) if the eccentricity of the hyperbola, x^{2} - y^{2} sec^{2} θ = 10 is 5 times the eccentricity of the ellipse, x^{2} sec^{2} θ + y^{2} = 5 then the length of the latus rectum of the ellipse, is:$ (1) $26$ (2) $\frac{2 5}{3}$ (3) $\frac{4 5}{3}$ (4) $30$ 96. $The locus of z satisfying \| z \| + \| z - 1 \| = 3 is$ (1) $a circle$ (2) $a pair of straight lines$ (3) $an ellipse$ (4) $a parabela$ 97. $If P is any point on the cllipse \frac{x ^{2}}{36} + \frac{y ^{2}}{16} = 1, and S and S^{’} are the foci, then PS + P S^{’} =$ (1) $4$ (2) $8$ (3) $10$ (4) $12$ 98. $Tangents are drawn to the ellipse \frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1 at all the four ends of its latusrectum. Then the area (in sq. units) of the quadrilateral formed by these tangents is$ (1) $\frac{125}{6}$ (2) $\frac{250}{3}$ (3) $\frac{80}{3}$ (4) $\frac{260}{3}$ 99. $Let P be a point on the ellipse \frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1 . Let the line passing through P and parallel to y -axis meet the circle x^{2} + y^{2} = 9 at point Q such that P and Q are on the same side of the x -axis. Then, the eccentricity of the locus of the point R on PQ such that PR : RQ = 4 : 3 as P moves on the ellipse, is :$ (1) $\frac{11}{19}$ (2) $\frac{13}{21}$ (3) $\frac{139}{23}$ (4) $\frac{13}{7}$ 100. $The acute angle between the pair of tangents drawn to the ellipse 2 x^{2} + 3 y^{2} = 5 from the point (1,3) is$ (1) $tan^{- 1} (\frac{16}{7 5})$ (2) $tan^{- 1} (\frac{24}{7 5})$ (3) $tan^{- 1} (\frac{32}{7 5})$ (4) $tan^{- 1} (\frac{3 + 8 5}{35})$ 101. $Let the eccentricity of an ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b, be \frac{1}{4} . If this ellipse passes through the point (- 4 \frac{2}{5}, 3), then a^{2} + b^{2} is equal to :$ (1) $29$ (2) $31$ (3) $32$ (4) $34$ 102. $Let a circle of radius 4 be concentric to the ellipse 15 x^{2} + 19 y^{2} = 285. Then the common tangents are inclined to the minor axis of the ellipse at the angle$ (1) $\frac{π}{3}$ (2) $\frac{π}{4}$ (3) $\frac{π}{6}$ (4) $\frac{π}{12}$ 103. $If tangents are drawn to the ellipse x^{2} + 2 y^{2} = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve:$ (1) $\frac{1}{4 x ^{2}} + \frac{1}{2 y ^{2}} = 1$ (2) $\frac{x ^{2}}{4} + \frac{y ^{2}}{2} = 1$ (3) $\frac{1}{2 x ^{2}} + \frac{1}{4 y ^{2}} = 1$ (4) $\frac{x ^{2}}{2} + \frac{y ^{2}}{4} = 1$ 104. $If the curve x^{2} + 2 y^{2} = 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by theline segment PQ at the origin is:$ (1) $\frac{π}{2} + tan^{- 1} (\frac{1}{4})$ (2) $\frac{π}{2} - tan^{- 1} (\frac{1}{4})$ (3) $\frac{π}{2} + tan^{- 1} (\frac{1}{3})$ (4) $\frac{π}{2} - tan^{- 1} (\frac{1}{3})$ 105. $A focus of an ellipse at the origin. The directrix is the line x = 4 and eccentricity is \frac{1}{2}, then the length of the semi major axis is$ (1) $\frac{8}{3}$ (2) $\frac{2}{3}$ (3) $\frac{4}{3}$ (4) $\frac{5}{3}$ 106. $The point (1, 3) with respect to the ellipse 4 x^{2} + 9 y^{2} - 16 x - 54 y + 61 = 0 lies$ (1) $outside the ellipse$ (2) $on the ellipse$ (3) $on the minor axis$ (4) $on the major axis$ 107. $The line y = x +1 meets the ellipse \frac{x ^{2}}{4} + \frac{y ^{2}}{2} = 1 at two points P and Q. If r is the radius of the circle with PQ as diameter then (3 r)^{2} is equal to$ (1) $20$ (2) $12$ (3) $11$ (4) $8$ 108. $If the midpoint of a chord of the ellipse \frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1 is (2, 4/3), and the length of the chord is \frac{2 α}{3}, then α is$ (1) $20$ (2) $22$ (3) $18$ (4) $26$ 109. $Let S and S^{’} be the foci of an ellipse and B be one end of its minor axis. If SB S^{’} is a isosceles right angled triangle then the eccentricity of the ellipse is$ (1) $\frac{1}{2}$ (2) $\frac{1}{2}$ (3) $\frac{3}{2}$ (4) $\frac{1}{3}$ 110. $The product of the lengths of the perpendiculars drawn from the two foci of the ellipse \frac{x ^{2}}{9} + \frac{y ^{2}}{25} = 1 to the tangent at any point on the ellipse is$ (1) $6$ (2) $7$ (3) $8$ (4) $9$ 111. $The ellipse x2 + 4y2 = 4 is inscribed in a rectangle alingent with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point (4, 0). Then the equation of the ellipse is :$ (1) $x2 + 12y2 = 16$ (2) $4x2 + 48y2 = 48$ (3) $4x2 + 64y2 = 48$ (4) $x2 + 16y2 = 16$ 112. $A hyperbola passes through the point P(2, 3) and has foci at (\pm2, 0). Then the tangent to this hyperbola at P also passes through the point :$ (1) $(3 2, 2 3)$ (2) $(2 2, 3 3)$ (3) $(3, 2)$ (4) $(- 2, - 3)$ 113. $Let S and S^{’} be the foci of an ellipse and B be any one of the extremities of its minor axis. If Δ S^{’} BS is a right angled triangle with right angle at B and area (Δ S^{’} BS) = 8 sq units, hen the length of a latus rectum of the ellipse is:$ (1) $4$ (2) $22$ (3) $42$ (4) $2$ 114. $Let E_{1} = \frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1 and E_{2} = \frac{x ^{2}}{d ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 be two ellipses and R be a rectangle with sides parallel to the coordinate axes. Let E_{1} be inscribed ellipse in R and E_{2} be circumscribed ellipse on R. If E_{2} passes through (0, 4) then$ (1) $a = 4, b = 23$ (2) $a = 12, b = 16$ (3) $a = 16, b = 16$ (4) $a = 23, b = 4$ 115. $Equation of the ellipse whose axes are the axes of coordinates and which passes through the point (-3, 1) and has eccentricity \frac{2}{5} is :$ (1) $3 x^{2} + 5 y^{2} - 32 = 0$ (2) $5 x^{2} + 3 y^{2} - 48 = 0$ (3) $3 x^{2} + 5 y^{2} - 15 = 0$ (4) $5 x^{2} + 3 y^{2} - 32 = 0$ 116. $Let L be a common tangent line to the curves 4 x^{2} + 9 y^{2} = 36 and (2 x)^{2} + (2 y)^{2} = 31 . Then the square ofthe slope of the line L is$ 117. $Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3,-4), one focus at (4,-4) and one vertex at (5,-4). If mx - y = 4, m > 0 is a tangent to the ellipse E, then the value of 5 m^{2} is equal to$ 118. $If the length of the latus rectum of the ellipse x^{2} + 4 y^{2} + 2x + 8y - λ = 0 is 4, and l is the length of its major axis, then λ + l is equal to_______.$ 119. $Let the common tangents to the curves 4(x^{2} + y^{2}) = 9 and y^{2} = 4x intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and l respectively denote the eccentricity and the length of the latus rectum of this ellipse, then \frac{l}{e ^{2}} is equal to___________.$ 120. $Let E_{1} : \frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1 be an ellipse. Ellipses E_{1}'s are constructed such that their centres and eccentricities are same as that of E_{1}, and the length of minor axis of E_{i} is the length of major axis of E_{i + 1} (i \geq 1) . If A_{i} is the area of the ellipse E_{i}, then \frac{5}{π} (\sum A_{i}), is equal to$