Hyperbola Questions with Solutions

8 questions1 easy4 medium3 hard

A hyperbola is the set of points where the difference of the distances from two fixed points (the foci) is constant. Its standard form is x^2/a^2 - y^2/b^2 = 1, with eccentricity given by b^2 = a^2(e^2 - 1), where e > 1. The foci are at (ae, 0) and (-ae, 0), and the asymptotes are the lines y = (b/a)x and y = -(b/a)x. Exam questions test finding the foci, eccentricity, transverse and conjugate axes, latus rectum, asymptotes, and equations of tangents. A special case is the rectangular hyperbola, where a = b and e = sqrt 2. Many students confuse the hyperbola with the ellipse, so compare the signs in the equation and remember that the difference of focal distances equals 2a. Practise the hyperbola questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
Find the area of the triangle formed in the right half-plane by the lines $x - y = 0$ and $x + y = 0$ and a tangent to the hyperbola $x^{2} - y^{2} = a^{2}$ , where $a$ is a non-zero number.
A. $a^{2}$ B. $\frac{a ^{2}}{2}$ C. $2 a^{2}$ D. $4 a^{2}$
mediumAsked in NIMCET 2026View solution ›
Q2.
The curve for which the normal at any point $(x, y)$ and the line joining the origin to that point form an isosceles triangle with the $x$ -axis as the base is
A. An ellipseB. A rectangular hyperbolaC. A circleD. A square
hardAsked in MAH MCA CET 2024View solution ›
Q3.
The foci of the conic $x^{2} + 2 x - 4 y^{2} + 8 y - 7 = 0$ are
A. $(5 \pm 1, 1)$ B. $(- 1 \pm 5, 1)$ C. $(- 1, 1 \pm 5)$ D. $(1, - 1 \pm 5)$
mediumAsked in NIMCET 2023View solution ›
Q4.
The straight lines $\frac{x}{a} + \frac{y}{b} = k$ and $\frac{x}{a} - \frac{y}{b} = \frac{1}{k}$ (with $k \neq = 0$ ) meet on:
A. a parabolaB. an ellipseC. a hyperbolaD. a circle
mediumAsked in NIMCET 2009View solution ›
Q5.
The eccentricity of the hyperbola $\frac{x ^{2}}{16} - \frac{y ^{2}}{9} = 1$ is
A. $\frac{5}{4}$ B. $\frac{4}{5}$ C. $\frac{5}{3}$ D. $\frac{7}{4}$
easyView solution ›
Q6.
The locus of the intersection of the two lines $3 x - y = 4 k 3$ and $k (3 x + y) = 43$ , for different values of k, is a hyperbola. The eccentricity of the hyperbola is:
A. $1.5$ B. $3$ C. $2$ D. $\frac{3}{2}$
mediumAsked in NIMCET 2014View solution ›
Q7.
The equation of the common tangent with positive slope to the circle $x^{2} + y^{2} - 8 x = 0$ and the hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ is
A. $2 x - 5 y - 20 = 0$ B. $2 x - 5 y + 4 = 0$ C. $3 x - 4 y + 8 = 0$ D. $4 x - 3 y + 4 = 0$
hardAsked in NIMCET 2018View solution ›
Q8.
Let $F_{1}$ , $F_{2}$ be the foci of the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ , $a > 0$ , $b > 0$ , and let O be the origin. Let M be a point on the hyperbola above the x-axis and H be a point on $M F_{1}$ such that $M F_{2} ⊥ F_{1} F_{2}$ , $M F_{1} ⊥ O H$ and $∣ O H ∣ = λ ∣ O F_{2} ∣$ with $λ \in (\frac{2}{5}, \frac{3}{5})$ . Then the range of the eccentricity $e$ is
A. $(\frac{7}{3}, 2)$ B. $(2, 3)$ C. $(1, \frac{7}{3})$ D. $(3, 2)$
hardAsked in NIMCET 2025View solution ›