Hyperbola Questions - JEE

The equation of the transverse and conjugate axes of a hyperbola are respectively x + 2y - 3 = 0, 2x - y + 4 = 0 and their respective lengths are $\sqrt{2}$ and $\frac{2}{\sqrt{3}}$. The equation of the hyperbola is:

The length of sub-tangent to the hyperbola $x^2 - 4y^2 = 4$ corresponding to the normal having slope unity is $\dfrac{1}{\sqrt{k}}$, then the value of $k$ is:

Tangents are drawn to the hyperbola $4x^2 - y^2 = 36$ at the points $P$ and $Q$. If these tangents intersect at the point $T(0, 3)$ then the area (in sq. units) of $\Delta PTQ$ is

Equation of the hyperbola with eccentricity $\frac{3}{2}$ and foci at $(\pm 2, 0)$ is

The equation of the hyperbola whose conjugate axis is 5 and the distance between the foci is 13, is:

278. On a coordinate plane, ellipse $C_1: \dfrac{x^2}{a_1^2}+\dfrac{y^2}{b_1^2}=1$ ($a_1>b_1>0$) and hyperbola $C_2: \dfrac{x^2}{a_2^2}+\dfrac{y^2}{b_2^2}=1$ ($a_2, b_2>0$) has the same focus point $F_1, F_2$. Point $P$ is the intersection point of $C_1$ and $C_2$ in the first quadrant and $|F_1F_2|=2|PF_2|\cdot e_1$ is the eccentricity of $C_1$ and $e_2$ is the eccentricity of $C_2$. Find the range of $e_2-e_1$.

The locus of a point $P(\alpha, \beta)$ moving under the condition that the line $y = \alpha x + \beta$ is a tangent to the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ is

If the tangents drawn to the hyperbola $4y^2 = x^2 + 1$ intersect the co-ordinate axes at the distinct points $A$ and $B$, then the locus of the midpoint of $AB$ is

The line 3x - 4y = 5 is a tangent to the hyperbola x2 - 4y2 = 5. The point of contact is

Let \(0

The vertices of a hyperbola are at $(0, 0)$ and $(10, 0)$ and one of its foci is at $(18, 0)$. The equation of hyperbola is

Let $P$ be a point on the hyperbola $H:\dfrac{x^2}{9}-\dfrac{y^2}{4}=1$, in the first quadrant such that the area of triangle formed by $P$ and the two foci of $H$ is $2\sqrt{13}$. Then the square of the distance of $P$ from the origin is

Consider a hyperbola $H$ having centre at the origin and foci and the $x$-axis. Let $C_{1}$ be the circle touching the hyperbola $H$ and having the centre at the origin. Let $C_{2}$ be the circle touching the hyperbola $H$ at its vertex and having the centre at one of its foci. If areas (in sq. units) of $C_{1}$ and $C_{2}$ are $36 \pi$ and $4 \pi$, respectively, then the length (in units) of latus rectum of $H$ is

Find the length of the transverse axis of the rectangular hyperbola xy = 18.

The equation of hyperbola is \ $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Here, $2a = 4 \Rightarrow a = 2$. Since the line passes through $(4, 2)$, find the eccentricity of the hyperbola.

Tangent at P to rectangular hyperbola $xy = 2$ meets coordinate axes at A and B, then area of triangle OAB (where O is origin) is:

A hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $(\pm 2, 0)$. Then the tangent to this hyperbola at $P$ also passes through the point

Let $e$ be the eccentricity of a hyperbola and $f(e)$ be the eccentricity of its conjugate hyperbola, then $\underbrace{\int\int\int \cdots}_{n \text{ times}} f(e)\,de$ (integrated from 1 to 3) is equal to

Let the equation of hyperbola be $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$. It passes through $(4, 6)$ and the eccentricity is $2$. Find the equation of the tangent to the hyperbola at $(4, 6)$.

For $0 \lt \theta \lt \pi / 2$, if the eccentricity of the hyperbola $x^{2}-y^{2} \operatorname{cosec}^{2} \theta=5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^{2} \operatorname{cosec}^{2} \theta +y^{2}=5$, then the value of $\theta$ is:

Given: $4x^2 - 9y^2 = 36$. The equation of hyperbola is $\frac{x^2}{9} - \frac{y^2}{4} = 1$. A point $Q(3\sec\theta, 2\tan\theta)$ is on the hyperbola. The normal at $Q$ meets the co-ordinate axes at $A\left(\frac{13}{3}\sec\theta, 0\right)$ and $B\left(0, \frac{13}{2}\tan\theta\right)$. If $OABP$ is a parallelogram and coordinate of $P$ is $(h, k)$, then the locus of $P$ is:

Point of hyperbola \left(ct, \frac{c}{t}\right)\ lie on director circle x^2 + y^2 = a^2 + b^2\ of ellipse

Let P(3, 3) be a point on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. If the normal to it at P intersects the x-axis at (9, 0) and e is its eccentricity, then the ordered pair (a2, e2) is equal to:

The equation of hyperbola whose foci are (2, 4) and (-2, 4) and eccentricity is $\frac{4}{3}$, is

A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the $x$-axis. Then the eccentricity of the hyperbola is __________ (up to four decimal places).

Let one focus of the hyperbola. ${H}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}$ $=1$ be at $(\sqrt{10}, 0)$ and the corresponding directrix be $x=\frac{9}{\sqrt{10}}$. If $e$ and $l$ respectively are the eccentricity and the length of the latus rectum of H , then $9\left(e^{2}+l\right)$ is equal to:

If a hyperbola passes through the point $P(10, 16)$ and it has vertices at $(\pm 6, 0)$, then the equation of the normal to it at $P$ is

If the eccentricity of the standard hyperbola passing through the point $(4, 6)$ is 2, then the equation of the tangent to the hyperbola at $(4, 6)$ is:

The tangent at an extremity (in the first quadrant) of latus rectum of the hyperbola $\dfrac{x^2}{4} - \dfrac{y^2}{5} = 1$, meets x-axis and y-axis at $A$ and $B$ respectively. Then $(OA)^2 - (OB)^2$, where $O$ is the origin, equals

If the line y = mx + 7$\sqrt3$ is normal to the hyperbola $\frac{x^{2}}{24}-\frac{y^{2}}{18}=1$, then a value of m is

Consider a hyperbola H whose centre is at the origin and the line $x + y = 2$ touches it at point $(1, 1)$. The tangent $x + y = 2$ intersects the asymptotes of H at points A and B such that the length of segment $AB = 6\sqrt{2}$. Find the equation of the pair of directrices of H.

The equation of a hyperbola is $16x^2 - 9y^2 = 144$. If one of its directrices is $5x + 9 = 0$, find the corresponding focus.

Equation of hyperbola with respect to x-y system referring to transverse axis as x-axis and conjugate axis as y-axis respectively is:

Given: $4y^2 = x^2 + 1$. So, $\left(\tan\theta, \frac{1}{2}\sec\theta\right)$ lies on hyperbola. The tangent at this point meets the co-ordinate axes at points $A$ and $B$. If the mid-point of $AB$ is $(h, k)$, then the locus of the mid-point is:

The circle $x^2 + y^2 - 8x = 0$ and hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ intersect at the points A and B. Find the equation of a common tangent with positive slope to the circle as well as to the hyperbola.

If the foci of a hyperbola are same as that of the ellipse $\dfrac{x^2}{9}+\dfrac{y^2}{25}=1$ and the eccentricity of the hyperbola is $\dfrac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2},\dfrac{14}{3}\sqrt{\dfrac{2}{5}}\right)$ on the hyperbola is equal to

Consider an ellipse $\frac{x^2}{36} + \frac{y^2}{18} = 1$. There is a hyperbola whose one asymptote is the major axis of the given ellipse. If the eccentricity of the given ellipse and hyperbola are reciprocal to each other, both have the same centre, and both touch each other in the first and third quadrants, find the focus of the hyperbola.

Consider an ellipse $\frac{x^2}{36} + \frac{y^2}{18} = 1$ and a hyperbola as described above. How many points in the x-y plane exist from where tangents can be drawn to the hyperbola?

Let $P(3\sec\theta, 2\tan\theta)$ and $Q(3\sec\phi, 2\tan\phi)$ where $\theta + \phi = \dfrac{\pi}{2}$, be two distinct points on the hyperbola $\dfrac{x^2}{9} - \dfrac{y^2}{4} = 1$. Then the ordinate of the point of intersection of the normals at $P$ and $Q$ is

If the normal to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ at any point $P(a\sec\theta, b\tan\theta)$ meets the transverse and conjugate axes in G and g respectively and if F is the foot of perpendicular to the normal at P from the centre C, then the value of $(PG)^2$ is:

(A) $3x - 4y = 4$

Let T and C respectively be the transverse and conjugate axes of the hyperbola $16x^2 - y^2 + 64x + 4y + 44 = 0$. Then the area of the region above the parabola $x^2 = y + 4$, below the transverse axis T and on the right of the conjugate axis C is:

The locus of the point of intersection of the straight lines, $tx - 2y - 3t = 0$; $x - 2ty + 3 = 0$ ($t \in \mathbb{R}$), is

If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the tangent to the hyperbola at (4, 6) is

If A and B are the points of intersection of the circle $x^{2}+y^{2}-8 x=0$ and the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ and a point P moves on the line $2 x-3 y+4=0$, then the centroid of $\triangle {PAB}$ lies on the line:

The line x cos α + y sin α = p touches the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, if a2 cos2 α − b2 sin2 α is equal to

If a normal to a rectangular hyperbola at a point cuts off intercepts $a_1, a_2$ on one axis and $b_1, b_2$ on the other axis then $a_1 a_2 + b_1 b_2 =$ ___.

Let the foci of a hyperbola coincide with the foci of the ellipse $\dfrac{x^2}{36}+\dfrac{y^2}{16}=1$. If the eccentricity of the hyperbola is 5, then the length of its latus rectum is:

Let $S$ be the focus of the hyperbola $\dfrac{x^2}{3}-\dfrac{y^2}{5}=1$, on the positive $x$-axis. Let $C$ be the circle with its centre at $A(\sqrt{6},\sqrt{5})$ and passing through the point $S$. If $O$ is the origin and $SAB$ is a diameter of $C$, then the square of the area of the triangle $OSB$ is equal to

Let $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $a > b$ and $H : \dfrac{x^2}{A^2} - \dfrac{y^2}{B^2} = 1$. Let the distance between the foci of $E$ and the foci of $H$ both be $2\sqrt{3}$. If $a - A = 2$, and the ratio of the eccentricities of $E$ and $H$ is $\tfrac{1}{3}$, then the sum of the lengths of their latus rectums is equal to

Hyperbola Questions (161)