Hyperbola Questions (161)

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

Locus of a point, whose chord of contact with respect to the circle \(x^2 + y^2 = 4\) is a tangent to the hyperbola \(xy = 1\) is a/an:

Let the circle C touch the line x - y + 1 = 0, have the centre on the positive x -axis, and cut off a chord of length 2 2 y along the line -3x + 2y = 1. Let H be the hyperbola , whose one of the foci is the centre of C 4 x - = 1 2 2 \sqrt13 \alpha \beta and the length of the transverse axis is the diameter of C . Then 2\alpha + 3\beta is equal to ______ 2 2

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

Let a tangent to the curve $y^2 = 24x$ meet the curve $xy = 2$ at the points A and B. Then the mid points of such line segments AB lie on a parabola with the

The length of the latus rectum and directrices of a hyperbola with eccentricity $e$ are 9 and $x=\pm\dfrac{4}{\sqrt{13}}$, respectively. Let the line $y-\sqrt{3}x+\sqrt{3}=0$ touch this hyperbola at $(x_0,y_0)$. If $m$ is the product of the focal distances of the point $(x_0,y_0)$, then $4e^2+m$ is equal to

Let $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, $a>b$ be an ellipse, whose eccentricity is $\dfrac{1}{\sqrt{2}}$ and the length of the latus rectum is $\sqrt{14}$. Then the square of the eccentricity of $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ is:

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

Let $e_1$ be the eccentricity of the hyperbola $\dfrac{x^2}{16}-\dfrac{y^2}{9}=1$ and $e_2$ be the eccentricity of the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, $a>b$, which passes through the foci of the hyperbola. If $e_1e_2=1$, then the length of the chord of the ellipse parallel to the $x$-axis and passing through $(0,2)$ is:

Let the latus rectum of the hyperbola $\dfrac{x^2}{9}-\dfrac{y^2}{b^2}=1$ subtend an angle of $\dfrac{\pi}{3}$ at the centre of the hyperbola. If $b^2$ is equal to $\dfrac{l}{m}(1+\sqrt{n})$, where $l$ and $m$ are co-prime numbers, then $l^2+m^2+n^2$ is equal to

Let the circle $C$ touch the line $x - y + 1 = 0$, have the centre on the positive $x$-axis, and cut off a chord of length $\dfrac{4}{\sqrt{13}}$ along the line $-3x + 2y = 1$. Let $H$ be the hyperbola $\dfrac{x^2}{\alpha^2} - \dfrac{y^2}{\beta^2} = 1$, whose one of the foci is the centre of $C$ and the length of the transverse axis is the diameter of $C$. Then $2\alpha^2 + 3\beta^2$ is equal to _____.

Let $A$ be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is $-3$. If the points $(x,y)$ satisfying $A^2+xA+yI=O$ lie on a hyperbola, whose length of semi major axis is $x$ and semi minor axis is $y$, eccentricity is $e$ and the length of the latus rectum is $l$, then $81(e^4+l^2)$ is equal to

Let $H_1 : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ and $H_2 : -\dfrac{x^2}{A^2} + \dfrac{y^2}{B^2} = 1$ be two hyperbolas having lengths of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\dfrac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to _____.

263. \(F_1, F_2\) are left and right focus points of the hyperbola \(C : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1\) \((a > 0, b > 0)\). Point \(O\) is the origin of the coordinate, \(M\) is an arbitrary point on \(C\) and above the \(x\)-axis. \(H\) is a point of \(MF_1\). Given that \(MF_2 \perp F_1F_2\), \(MF_1 \perp OH\), \(|OH| = \lambda|OF_2|\), where \(\lambda \in \left(\dfrac{1}{3}, \dfrac{1}{2}\right)\). Find the range of the eccentricity of the hyperbola \(C\).

Two dice are thrown independently. Let A be the event that the number appeared on the \(1^{\text {st }}\) die is less than the number appeared on the \(2^{\text {nd }}\) die, B be the event that the number appeared on the \(1^{\text {st }}\) die is even and that on the second die is odd, and C be the event that the number appeared on the \(1^{\text {st }}\) die is odd and that on the \(2^{\text {nd }}\) is even. Then:

General equation for normal to hyperbola \(\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1\) is given. If the normal is \(y = mx + 7\sqrt{3}\), find the slope \(m\).

If two tangents can be drawn to the different branches of the hyperbola \(x^2 - \dfrac{y^2}{4} = 1\) from the point \((\alpha,\, \alpha^2)\), then:

The equation of parabola is \(y^2 = 12x\) and the equation of hyperbola is \(8x^2 - y^2 = 8\). A common tangent to both curves is drawn. If the tangents \(y = 3x + 1\) and \(y = -3x - 1\) intersect at point P, and S, S' are foci of the hyperbola, then the ratio \(\dfrac{SP}{PS'}\) is:

If \(5x + 9 = 0\) is the directrix of the hyperbola \(16x^2 - 9y^2 = 144\), then its corresponding focus is:

If a directrix of a hyperbola centred at the origin and passing through the point \((4, -2\sqrt{3})\) is \(5x = 4\sqrt{5}\) and its eccentricity is \(e\), then

The length of transverse axis of the hyperbola 3x2 - 4y2 = 32 is

If \(P(x_1, y_1)\), \(Q(x_2, y_2)\), \(R(x_3, y_3)\) and \(S(x_4, y_4)\) are four concyclic points on the rectangular hyperbola \(xy = c^2\), then coordinates of the orthocentre of the \(\triangle PQR\) are

Let P be the point of intersection of the common tangents to the parabola \(y^2 = 12x\) and the hyperbola \(8x^2 - y^2 = 8\). If S and S' denote the foci of the hyperbola where S lies on the positive \(x\)-axis then P divides SS' in a ratio:

Let \(a\) and \(b\), respectively, be the semi-transverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the equation \(9e^2 - 18e + 5 = 0\). If \(S(5, 0)\) is a focus and \(5x = 9\) is the corresponding directrix of this hyperbola, then \(a^2 - b^2\) is equal to

If the line $\alpha x+2y=1$, where $\alpha\in\mathbb{R}$, does not meet the hyperbola $x^2-9y^2=9$, then a possible value of $\alpha$ is:

For some $\theta\in\left(0,\dfrac{\pi}{2}\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $x^2-y^2\sec^2\theta=8$ be $e_1$ and $l_1$, respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^2\sec^2\theta+y^2=6$ be $e_2$ and $l_2$, respectively. If $e_1^2=e_2^2(\sec^2\theta+1)$, then $\left(\dfrac{l_1 l_2}{e_1 e_2}\right)\tan^2\theta$ is equal to _____.

If a hyperbola passes through the point \(P(2, 3)\) and has foci at \((\pm 2, 0)\), then the tangent to this hyperbola at \(P\) also passes through the point

The point of intersection of two tangents to the hyperbola \(\frac{x^2}{25} - \frac{y^2}{144} = 1\), the product of whose slopes is \(c^2\), lies on the curve

All chords of a curve \(3x^2 - y^2 - 2x + 4y = 0\) which subtends a right angle at the origin passes through a fixed point, which is:

If two tangents can be drawn to the different branches of the hyperbola \(x^2 - \dfrac{y^2}{4} = 1\) from the point \((\alpha, \alpha^2)\), then:

The equation of the transverse and conjugate axis of the hyperbola 16x2 − y2 + 64x + 4y + 44 = 0 are:

The vertices of the hyperbola 9x2 - 16y2 - 36x + 96y - 252 = 0 are

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

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 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 number of points in the x-y plane from where perpendicular tangents can be drawn to the hyperbola.

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 tangent to H at the point \(\left(-1, \frac{7}{2}\right)\) on it.

Find the equation of the tangent to the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ that is also tangent to the circle with diameter $AB$, where $A$ and $B$ are specific points on the hyperbola.

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:

Let the domain of the function $f(x)=\log_3\log_5\log_7(9x-x^2-13)$ be the interval $(m,n)$. Let the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ have eccentricity $\dfrac{n}{3}$ and the length of the latus rectum $\dfrac{8m}{3}$. Then $b^2-a^2$ is equal to:

Let $P(10,2\sqrt{15})$ be a point on the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$, whose foci are $S$ and $S'$. If the length of its latus rectum is 8, then the square of the area of $\triangle PSS'$ is equal to:

A hyperbola has transverse axis of length \(2\sin\theta\) and is confocal with the ellipse \(3x^2 + 4y^2 = 12\). Then its equation is: