Hyperbola Questions - JEE

The tangent to the hyperbola $x^2 - 3y^2 = 3$ at the point $(\sqrt{3}, 0)$ when associated with two asymptotes constitutes:

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 $H$.

For the hyperbola $H$ as described above, find the equation of the tangent to $H$ at the point $\left(-1, \frac{7}{2}\right)$ on it.

Consider a branch of the hyperbola, $x^2 - 2y^2 - 2\sqrt{2}x - 4\sqrt{2}y - 6 = 0$ with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$, then the area of the triangle $ABC$ 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 $(PF)^2$ 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 $|PF| \cdot |PG|$ is equal to:

For a hyperbola $9x^2 - 5y^2 = 1$, if P lies on the hyperbola at $(r_1\cos\theta, r_1\sin\theta)$ and Q lies on the hyperbola at $(r_2\cos(\theta+90°), r_2\sin(\theta+90°))$, find $\frac{1}{r_1^2} + \frac{1}{r_2^2}$.

The length of the transverse axis of a hyperbola is 7 and it passes through the point (5, 2). 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:

A hyperbola whose transverse axis is along the major axis of the conic, $\dfrac{x^2}{3} + \dfrac{y^2}{4} = 4$ and has vertices at the foci of this conic. If the eccentricity of the hyperbola is 3/2, then which of the following points does NOT lie on it?

Let the equation of hyperbola be $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$. It passes through the point $(4, -2\sqrt{3})$ and the directrix is $5x = 4\sqrt{5}$. Find the eccentricity of the hyperbola.

The equation of a tangent to the hyperbola $x^2 - 2y^2 = 18$ which is perpendicular to the line $x - y = 0$ is

For the hyperbola $\dfrac{x^2}{\cos^2\alpha} - \dfrac{y^2}{\sin^2\alpha} = 1$, which of the following remains constant when $\alpha$ varies?

Given the equation of hyperbola $\dfrac{x^2}{\cos^2\theta} - \dfrac{y^2}{\sin^2\theta} = 1$ whose eccentricity $e > 2$. Then the latus rectum lies in the interval:

A normal to the hyperbola, $4x^2 - 9y^2 = 36$ meets the co-ordinate axes $x$ and $y$ at $A$ and $B$, respectively. If the parallelogram $OABP$ ($O$ being the origin) is formed, then the locus of $P$ is

If $S_1$ and $S_2$ are the foci of the hyperbola whose transverse axis length is 4 and conjugate axis length is 6, $S_3$ and $S_4$ are the foci of the conjugate hyperbola, then the area of the quadrilateral $S_1 S_3 S_2 S_4$ is $k$. Find $k/4$.

For a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, if $2ae = 16$ and $e = \frac{\sqrt{5}}{2}$, find $a^2$.

The equation of the tangent at the point '$\theta$' on the hyperbola $4x^2 - 3y^2 = 12$ is ___ where $\theta = \pi/4$.

If a circle circle by assuming a chord parallel to the transverse axis of hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ as diameter always passes through $(2,0)$, then

Equation of tangent to hyperbola \frac{xh}{a^2} - \frac{yk}{b^2} = 1\ at point P(h,k)\ touches the parabola at point (at^2, 2at)\

Let $x^2 + y^2 = 4r^2$ and $xy = 1$ intersects at $A$ and $B$ in first quadrant. If $AB = \sqrt{14}$ units, then the value of $|r|$ is

Let the chord $x\cos\alpha + y\sin\alpha = p$ of the hyperbola $\frac{x^2}{16} - \frac{y^2}{18} = 1$ subtends a right angle at the centre. Let diameter of the circle, concentric with the hyperbola, to which the given chord is a tangent is $d$, then $\frac{d}{4}$ is equal to:

For the hyperbola $\frac{x^2}{9} - \frac{y^2}{3} = 1$ the incorrect statement is:

The equation of a hyperbola is $ \dfrac{x^2}{\cos^2\theta} - \dfrac{y^2}{\sin^2\theta} = 1 $ with eccentricity $ e > 2 $. The latus rectum of this hyperbola belongs to which interval?(1) $(3, \infty)$ (2) $(1, 3)$ (3) $(2, 4)$ (4) $(1, 2)$

If x, y ∈ ℝ satisfy the equation \[\frac{(x+4)^2}{4}\] − \[\frac{y^2}{9}\] = 1, then the difference between the largest and smallest value of the expression \[\frac{x^2}{4}\] + \[\frac{y^2}{9}\] is

The eccentricity of the hyperbola $16x^2 - 9y^2 = 144$ is

Let any double ordinate $PNP'$ of the hyperbola $\dfrac{x^2}{25} - \dfrac{y^2}{16} = 1$ be produced on both sides to meet the asymptotes in $Q$ and $Q'$, then $\dfrac{PQ \cdot P'Q}{5}$ is equal to

Consider a hyperbola $H$ having centre at the origin and foci on 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

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E:\dfrac{(x-1)^2}{100}+\dfrac{(y-1)^2}{75}=1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3\alpha^2+2\beta^2$ is equal to

The foci of a hyperbola are $(\pm2,0)$ and its eccentricity is $\dfrac{3}{2}$. A tangent perpendicular to $2x+3y=6$ is drawn at a point in the first quadrant. If $a$ and $b$ are the $x$- and $y$-intercepts of the tangent, then $|6a|+|5b|$ is equal to

Let $m_1$ and $m_2$ be the slopes of the tangents drawn from the point $P(4,1)$ to the hyperbola $H:\ \dfrac{y^2}{25}-\dfrac{x^2}{16}=1$. If $Q$ is the point from which tangents with slopes $|m_1|$ and $|m_2|$ make positive $x$-intercepts $\alpha$ and $\beta$, then $\dfrac{(PQ)^2}{\alpha\beta}$ is equal to

Let the foci of a hyperbola be (1, 14) and (1, -12). If it passes through the point (1, 6), then the length of its latus-rectum is :

Let H : 2 y 2 y and H : - be two hyperbolas having length of latus rectums 15\sqrt2 and x x 1 - = 1 2 + = 1 2 2 2 2 a b A B 12\sqrt5 respectively. Let their ecentricities be e = \sqrt and e respectively. If the product of the lengths of their 1 5 2 2 transverse axes is 100\sqrt10, then 25e is equal to ________. 2 2 2

Let the foci and length of the latus rectum of an ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, $a>b$ be $(\pm5,0)$ and $\sqrt{50}$, respectively. Then the square of the eccentricity of the hyperbola $\dfrac{x^2}{b^2}-\dfrac{y^2}{a^2b^2}=1$ equals

Let the foci of a hyperbola be $(1, 14)$ and $(1, -12)$. If it passes through the point $(1, 6)$, then the length of its latus-rectum is

Let $H:\dfrac{-x^2}{a^2}+\dfrac{y^2}{b^2}=1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the point $(\alpha,6)$, $\alpha>0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha,6)$, then $\alpha^2+\beta$ is equal to

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

The locus of the point of intersection of the lines, $\sqrt{2}x - y + 4\sqrt{2}k = 0$ and $\sqrt{2}kx + ky - 4\sqrt{2} = 0$ ($k$ is any non-zero real parameter), is

Director circle is the set of points from where drawn tangents are perpendicular, in this case $x^2 + y^2 = a^2 - b^2$ (equation of director circle) i.e., $x^2 + y^2 = -9$ is not a real circle, so there is no point from where perpendicular tangents can be drawn.

Given hyperbola is $3x^2 - 2y^2 = 6$ or $\frac{x^2}{2} - \frac{y^2}{3} = 1$. Slope form of tangent is $y = mx \pm \sqrt{a^2m^2 - b^2}$ or $(mx - y)^2 = a^2m^2 - b^2$. Tangent from the point $(\alpha, \beta)$ is given by, $(\beta - m\alpha)^2 = 2m^2 - 3$, i.e., $m^2(\alpha^2 - 2) - 2\alpha m\beta + \beta^2 + 3 = 0$, so $m_1m_2 = \frac{\beta^2 + 3}{\alpha^2 - 2} = \tan\theta \tan\phi$.

A conic passes through the point (2, 4) and is such that the segment of any of its tangents at any point contained between the co-ordinate axes is bisected at the point of tangency. Then the foci of the conic are:

Two tangents, one from $A(2,1)$ and other from $B(-2,1)$, are drawn to the hyperbola $\dfrac{x^2}{4}-y^2=1$. A circle which touches these two tangents and two asymptotes of the hyperbola has its centre at $(a,\lambda)$ where $\lambda>0$. The least area of the pentagon in which this circle is inscribed (two sides are asymptotes, two sides are the tangents) is:

The eccentricity of a hyperbola passing through the points (3, 0), (3$\sqrt 2$, 2) will be:

A point of the curve $\frac{x^{2}}{A^{2}}-\frac{y^{2}}{B^{2}}=1$ is

Find the equation of axis of the given hyperbola $\frac{x^{2}}{3}-\frac{y^{2}}{2}=1$ which is equally inclined to the axes?

If the tangent and normal at a point on rectangular hyperbola cut-off intercept $a_1, a_2$ on x-axis and $b_1, b_2$ on the y-axis, then $a_1a_2 + b_1b_2$ is equal to:

Let $PQ$ be a chord of the hyperbola $\dfrac{x^2}{4}-\dfrac{y^2}{b^2}=1$, perpendicular to the $x$-axis such that $OPQ$ is an equilateral triangle, $O$ being the centre of the hyperbola. If the eccentricity of the hyperbola is $\sqrt{3}$, then the area of the triangle $OPQ$ is

Let $P(x_0, y_0)$ be the point on the hyperbola $3x^2 - 4y^2 = 36$, which is nearest to the line $3x + 2y = 1$. Then $\sqrt{2}(y_0 - x_0)$ is equal to:

If (0, $\pm$4) and (0, $\pm$2) be the foci and vertices of a hyperbola, then its equation is

Hyperbola Questions (161)