Ellipse Questions with Solutions

14 questions4 easy10 medium

An ellipse is the set of points whose distances from two fixed points (the foci) add up to a constant. Its standard form is x^2/a^2 + y^2/b^2 = 1, where a > b, and its eccentricity e satisfies b^2 = a^2(1 - e^2), with 0 < e < 1. The foci are at (ae, 0) and (-ae, 0) and the directrices are x = a/e and x = -a/e. Exam questions test finding the foci, eccentricity, length of the major and minor axes, latus rectum (2b^2/a), focal distances, and equations of tangents and normals. A useful fact is that the sum of the focal distances of any point on the ellipse equals the length of the major axis, 2a. Always check whether the major axis lies along the x-axis or the y-axis before applying formulas. Practise the ellipse questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
The number of values of $C$ such that the straight line $y = 4 x + C$ touches the curve $x^{2} + 4 y^{2} = 4$ is
A. 2B. 0C. 1D. Infinite
mediumAsked in MAH MCA CET 2023View solution ›
Q2.
The eccentricity of an ellipse whose centre is at the origin is $\frac{1}{2}$ . If one of its directrices is $x = - 4$ , find the equation of the normal to the ellipse at the point $(1, \frac{3}{2})$ .
A. 4x + 2y = 7B. $6 x - 3 y = \frac{3}{2}$ C. $6 x + 3 y = \frac{21}{2}$ D. 4x − 2y = 1
mediumAsked in NIMCET 2026View solution ›
Q3.
The equation of the ellipse with major axis along the x-axis and passing through the points $(4, 3)$ and $(- 1, 4)$ is
A. $15 x^{2} + 7 y^{2} = 247$ B. $7 x^{2} + 15 y^{2} = 247$ C. $16 x^{2} + 9 y^{2} = 247$ D. $9 x^{2} + 16 y^{2} = 247$
mediumAsked in NIMCET 2012View solution ›
Q4.
If any tangent to the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ intercepts equal lengths $l$ on both the axes, then $l =$
A. $a^{2} + b^{2}$ B. $a^{2} + b^{2}$ C. $(a^{2} + b^{2})^{2}$ D. None of these
mediumAsked in NIMCET 2010View solution ›
Q5.
The tangent to an ellipse $x^{2} + 16 y^{2} = 16$ and making angle 60° with X-axis is:
A. $x - 3 y + 7 = 0$ B. $3 x - y + 8 = 0$ C. $3 x - y + 7 = 0$ D. $x + 3 y - 7 = 0$
mediumAsked in NIMCET 2020View solution ›
Q6.
The locus of the point of intersection of tangents to the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ which meet at right angles is
A. a circleB. a parabolaC. an ellipseD. a hyperbola
mediumAsked in NIMCET 2021View solution ›
Q7.
The eccentric angles of the extremities of the latus rectum of the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ are given by
A. $t an^{- 1} (\pm \frac{a e}{b})$ B. $t an^{- 1} (\pm \frac{b e}{e})$ C. $t an^{- 1} (\pm \frac{b}{a e})$ D. $t an^{- 1} (\pm \frac{a}{b e})$
mediumAsked in NIMCET 2021View solution ›
Q8.
The eccentricity of an ellipse with its centre at the origin is $\frac{1}{3}$ . If one of the directrices is $x = 9$ , then the equation of the ellipse is
A. $9 x^{2} + 8 y^{2} = 72$ B. $8 x^{2} + 9 y^{2} = 72$ C. $8 x^{2} + 7 y^{2} = 56$ D. $7 x^{2} + 8 y^{2} = 56$
mediumAsked in NIMCET 2022View solution ›
Q9.
An equation of a tangent from the point $(3, - 1)$ to the ellipse $2 x^{2} + 9 y^{2} = 3$ is
A. $2 x - 3 y - 3 = 0$ B. $2 x + 3 y - 3 = 0$ C. $2 x + y - 3 = 0$ D. None of these
mediumAsked in NIMCET 2019View solution ›
Q10.
If $(4, 3)$ and $(12, 5)$ are the two foci of an ellipse passing through the origin, then the eccentricity of the ellipse is
A. $\frac{13}{9}$ B. $\frac{13}{18}$ C. $\frac{17}{18}$ D. $\frac{17}{9}$
mediumAsked in NIMCET 2024View solution ›