Hyperbola

Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2-a x+2=0$. Let the sets $\left\{a \in \mathbb{R}: e_1\right.$ and $e_2$ are the eccentricities of hyperbolas $\}=(\alpha, \beta)$, and $\left\{a \in \mathbb{R}: e_1\right.$ and $...

If the eccentricity $e$ of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, passing through $(6,4 \sqrt{3})$, satisfies $15\left(e^2+1\right)=34 e$, then the length of the latus rectum of the hyperbola $\frac{x^2}{b^2}-\frac{y^2}{2\left(a...

The eccentricity of an ellipse $E$ with centre at the origin $O$ is $\frac{\sqrt{3}}{2}$ and its directrices are $x= \pm \frac{4 \sqrt{6}}{3}$. Let $\mathrm{H}: \frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$ be a hyperbola whose ecce...

Let the eccentricity e of a hyperbola satisfy the equation $6 \mathrm{e}^2-11 \mathrm{e}+3=0$. If the foci of the hyperbola are $(3,5)$ and $(3,-4)$, then the length of its latus rectum is :

Let $\mathrm{H}: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is $\frac{8}{3}$. If the line $x=\alpha$ intersects the hyperbola H at the points A and...

Let O be the origin, and P and Q be two points on the rectangular hyperbola $xy = 12$ such that the midpoint of the line segment PQ is $\left( \frac{1}{2}, -\frac{1}{2} \right)$. Then the area of the triangle OPQ equals :

For some $\theta \in\left(0, \frac{\pi}{2}\right)$, 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 r...

Let the ellipse $E: \frac{x^2}{144} + \frac{y^2}{169} = 1$ and the hyperbola $H: \frac{x^2}{16} - \frac{y^2}{\lambda^2} = -1$ have the same foci. If $e$ and $L$
respectively denote the eccentricity and the length of the latus rectum of $H$...

Let the domain of the function $f(x)=\log _3 \log _5 \log _7\left(9 x-x^2-13\right)$ be the interval $(\mathrm{m}, \mathrm{n})$. Let the hyperbola $\frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$ have eccentricity $\frac{\mathrm{n}}{3...

Let PQ be a chord of the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$, 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 a...

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

Let $\mathrm{P}(10,2 \sqrt{15})$ be a point on the hyperbola $\frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$, whose foci are S and $\mathrm{S}^{\prime}$. If the length of its latus rectum is 8 , then the square of the area of $\Delta...

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

Consider the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ having one of its focus at $\mathrm{P}(-3,0)$. If the latus ractum through its other focus subtends a right angle at P and $a^2 b^2=\alpha \sqrt{2}-\beta, \alpha, \beta \in \mathbb{...

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be $2 a$ and $2 b$, respectively, and one focus and the corresponding directrix of this hyperbola be $(-5,0)$ and $5 x+9=0$, respectively. If the product o...

Let e1 and e2 be the eccentricities of the ellipse $\frac{x^2}{b^2} + \frac{y^2}{25} = 1$ and the hyperbola $\frac{x^2}{16} - \frac{y^2}{b^2} = 1$, respectively. If b < 5 and e1e2 = 1, then the eccentricity of the ellipse having its axes al...

Let the sum of the focal distances of the point $\mathrm{P}(4,3)$ on the hyperbola $\mathrm{H}: \frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$ be $8 \sqrt{\frac{5}{3}}$. If for H , the length of the latus rectum is $l$ and the produc...

Let the product of the focal distances of the point $\mathbf{P}(4,2 \sqrt{3})$ on the hyperbola $\mathrm{H}: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be 32 . Let the length of the conjugate axis of H be $p$ and the length of its latus rectum be $...

If the equation of the hyperbola with foci $(4,2)$ and $(8,2)$ is $3 x^2-y^2-\alpha x+\beta y+\gamma=0$, then $\alpha+\beta+\gamma$ is equal to__________.

Let one focus of the hyperbola $\mathrm{H}: \frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$ 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 le...

Let $\mathrm{H}_1: \frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$ and $\mathrm{H}_2:-\frac{x^2}{\mathrm{~A}^2}+\frac{y^2}{\mathrm{~B}^2}=1$ be two hyperbolas having length of latus rectums $15 \sqrt{2}$ and $12 \sqrt{5}$ respectively...

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 the foci of a hyperbola \(H\) coincide with the foci of the ellipse \(E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1\) and the eccentricity of the hyperbola \(H\) be the reciprocal of the eccentricity of the ellipse \(E\). If the length of...

Let \(H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1\) 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 ...

Let \(\mathrm{S}\) be the focus of the hyperbola \(\frac{x^2}{3}-\frac{y^2}{5}=1\), on the positive \(x\)-axis. Let \(\mathrm{C}\) be the circle with its centre at \(\mathrm{A}(\sqrt{6}, \sqrt{5})\) and passing through the point $$\mathrm{S...

The length of the latus rectum and directrices of hyperbola with eccentricity e are 9 and \(x= \pm \frac{4}{\sqrt{3}}\), respectively. Let the line \(y-\sqrt{3} x+\sqrt{3}=0\) touch this hyperbola at \(\left(x_0, y_0\right)\). If $$\mathrm{...

Consider a hyperbola \(\mathrm{H}\) having centre at the origin and foci on the \(\mathrm{x}\)-axis. Let \(\mathrm{C}_1\) be the circle touching the hyperbola \(\mathrm{H}\) and having the centre at the origin. Let \(\mathrm{C}_2\) be the c...

Let the foci and length of the latus rectum of an ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b b e( \pm 5,0)\) and \(\sqrt{50}\), respectively. Then, the square of the eccentricity of the hyperbola $$\frac{x^2}{b^2}-\frac{y^2}{a^2 b^2}=...

If the foci of a hyperbola are same as that of the ellipse \(\frac{x^2}{9}+\frac{y^2}{25}=1\) and the eccentricity of the hyperbola is \(\frac{15}{8}\) times the eccentricity of the ellipse, then the smaller focal distance of the point $$\l...

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

Let \(P\) be a point on the hyperbola \(H: \frac{x^2}{9}-\frac{y^2}{4}=1\), 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 th...

Let \(e_1\) be the eccentricity of the hyperbola \(\frac{x^2}{16}-\frac{y^2}{9}=1\) and \(e_2\) be the eccentricity of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \mathrm{a} > \mathrm{b}\), which passes through the foci of the hyperbol...

For $0<\theta<\pi / 2$, 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 :

Let the eccentricity of an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is reciprocal to that of the hyperbola \(2 x^{2}-2 y^{2}=1\). If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectu...

Let $\mathrm{H}$ be the hyperbola, whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is :

The vertices of a hyperbola H are (\(\pm\) 6, 0) and its eccentricity is \({{\sqrt 5 } \over 2}\). Let N be the normal to H at a point in the first quadrant and parallel to the line \(\sqrt 2 x + y = 2\sqrt 2\). If d is the length of the l...

Let T and C respectively be the transverse and conjugate axes of the hyperbola \(16{x^2} - {y^2} + 64x + 4y + 44 = 0\). Then the area of the region above the parabola \({x^2} = y + 4\), below the transverse axis T and on the right of the co...

Let \(\mathrm{P}\left(x_{0}, y_{0}\right)\) be the point on the hyperbola \(3 x^{2}-4 y^{2}=36\), which is nearest to the line \(3 x+2 y=1\). Then \(\sqrt{2}\left(y_{0}-x_{0}\right)\) is equal to :

Let \(m_{1}\) and \(m_{2}\) be the slopes of the tangents drawn from the point \(\mathrm{P}(4,1)\) to the hyperbola \(H: \frac{y^{2}}{25}-\frac{x^{2}}{16}=1\). If \(\mathrm{Q}\) is the point from which the tangents drawn to \(\mathrm{H}\) h...

The foci of a hyperbola are \(( \pm 2,0)\) and its eccentricity is \(\frac{3}{2}\). A tangent, perpendicular to the line \(2 x+3 y=6\), is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the...

Let \(\mathrm{H}_{\mathrm{n}}: \frac{x^{2}}{1+n}-\frac{y^{2}}{3+n}=1, n \in N\). Let \(\mathrm{k}\) be the smallest even value of \(\mathrm{n}\) such that the eccentricity of \(\mathrm{H}_{\mathrm{k}}\) is a rational number. If \(l\) is the...

Let R be a rectangle given by the lines \(x=0, x=2, y=0\) and \(y=5\). Let A\((\alpha,0)\) and B\((0,\beta),\alpha\in[0,2]\) and \(\beta\in[0,5]\), be such that the line segment AB divides the area of the rectangle R in the ratio 4 : 1. The...

Let \(H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\), a > 0, b > 0, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is \(4(2\sqrt 2 + \sqrt {14} )\). If the eccentricity H is $${{\sqrt {11} }...

Let the eccentricity of the hyperbola \(H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) be \(\sqrt {{5 \over 2}}\) and length of its latus rectum be \(6\sqrt 2\). If \(y = 2x + c\) is a tangent to the hyperbola H, then the value...

Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\). Let e' and l' respectively be the eccentricity and length of the latus...

For the hyperbola \(\mathrm{H}: x^{2}-y^{2}=1\) and the ellipse \(\mathrm{E}: \frac{x^{2}}{\mathrm{a}^{2}}+\frac{y^{2}}{\mathrm{~b}^{2}}=1\), a \(>\mathrm{b}>0\), let the
(1) eccentricity of \(\mathrm{E}\) be reciprocal of the eccentricity ...

Let the hyperbola \(H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) pass through the point \((2 \sqrt{2},-2 \sqrt{2})\). A parabola is drawn whose focus is same as the focus of \(\mathrm{H}\) with positive abscissa and the directrix of the p...

An ellipse \(E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) passes through the vertices of the hyperbola \(H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1\). Let the major and minor axes of the ellipse \(E\) coincide with the transverse and conjug...

A common tangent \(\mathrm{T}\) to the curves \(\mathrm{C}_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) and \(C_{2}: \frac{x^{2}}{42}-\frac{y^{2}}{143}=1\) does not pass through the fourth quadrant. If \(\mathrm{T}\) touches \(\mathrm{C}_{1}\) ...

Let a line L1 be tangent to the hyperbola \({{{x^2}} \over {16}} - {{{y^2}} \over 4} = 1\) and let L2 be the line passing through the origin and perpendicular to L1. If the locus of the point of intersection of L1 and L2 is $${({x^2} + {y^2...