Hyperbola — JEE Main PYQs

Let the foci of 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:

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:

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

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

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

For some $\theta \in \left(0,\frac{\pi }{2}\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $x^2-{\mathrm{y}}^2{\sec }^2\theta =8$ be ${\mathrm{e}}_1$ and ${\ell }_1$, respectively, and let the eccentricity and the length of the latus r …

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

Let $\mathrm{H}:\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ 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 $\mathrm{x}=\alpha$ intersects the hyperbola H …

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 $(3,5)(3,-4)$, then the length of its latus rectum is:

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 $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be a hyperbola whose eccentricity is equal to the length of semi-major axis of …

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 $15(e^2+1)=34e$, then the length of the latus rectum of the hyperbola $\frac{x^2}{b^2}-\frac{y^2}{2(a^2+1)}=1$ is:

Let O be the origin, and P and Q be two points on the rectangular hyperbola $\mathrm{xy}=12$ such that the mid point of the line segment PQ is $\left(\frac{1}{2},-\frac{1}{2}\right)$. Then the area of the triangle OPQ 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 $H_1:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $H_2:-\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be ${\mathrm{e}}_1=\sqrt{\frac{5}{2}}$ and …

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

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 $\ell$ respectively are the eccentricity and the length of the latus rectum of $H$, then $9(e^2+\ell )$ is …

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

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

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

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

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 …

The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and $x=\pm \frac{4}{\sqrt{13}}$ 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 distance of the point …

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 of the focal distances of the point …

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

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 the transverse axis of H is $\alpha$ and the leng …

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 $b^2$ is equal to $\frac{l}{m}(1+\sqrt{n})$, where $l$ and $m$ are co-prime numbers, then $l^2+m^2+n^2$ is equal to

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 …

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

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

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

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

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

Let the tangent to the parabola ${\mathrm{y}}^2=12\mathrm{x}$ at the point $(3,\alpha )$ be perpendicular to the line $2\mathrm{x}+2\mathrm{y}=3$. Then the square of distance of the point $(6,-4)$ from the normal to the hyperbola …

Let $H_n=\frac{x^2}{1+n}-\frac{y^2}{3+n}=1,n\in N$. Let $k$ be the smallest even value of n such that the eccentricity of ${\mathrm{H}}_{\mathrm{k}}$ is a rational number. If $l$ is length of the latus rectum of ${\mathrm{H}}_{\mathrm{k}}$, then $21l$ is equal to

Let $\mathrm{P}\left({\mathrm{x}}_0,{\mathrm{y}}_0\right)$ be the point on the hyperbola $3{\mathrm{x}}^2-4y^2=36$, which is nearest to the line $3x+2y=1$. Then $\sqrt{2}\left({\mathrm{y}}_0-{\mathrm{x}}_0\right)$ is equal to :

Let 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

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

Let the equation of two diameters of a circle $x^2+y^2-2x+2fy+1=0$ be $2px-y=1$ and $2x+py=4p$. Then the slope $m\in (0,\infty )$ of the tangent to the hyperbola $3x^2-y^2=3$ passing through the centre of the circle is equal to ______.

If the line $x-1=0$, is a directrix of the hyperbola $k{x}^2-y^2=6$, then the hyperbola passes through the point

Let the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{7}=1$ and the hyperbola $\frac{x^2}{144}-\frac{y^2}{\alpha }=\frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is:-

Let the tangent drawn to the parabola $y^2=24x$ at the point $(\alpha ,\beta )$ is perpendicular to the line $2x+2y=5$. Then the normal to the hyperbola $\frac{x^2}{{\alpha }^2}-\frac{y^2}{{\beta }^2}=1$ at the point $(\alpha +4,\beta +4)$ does NOT pass through the point :

Let $\mathrm{a}>0,\mathrm{b}>0$. Let e and $\ell$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$. Let ${\mathrm{e}}^{\prime }$ and ${\ell }^{\prime }$ respectively the ec …

Let the eccentricity of the hyperbola $\mathrm{H}:\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ be $\sqrt{\frac{5}{2}}$ and length of its latus rectum be $6\sqrt{2}$, If $\mathrm{y}=2\mathrm{x}+\mathrm{c}$ is a tangent to the hyperbola …

The normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{9}=1$ at the point $(8,3\sqrt{3})$ on it passes through the point :

Let a line $L_1$ be tangent to the hyperbola $\frac{x^2}{16}-\frac{y^2}{4}=1$ and let $L_2$ be the line passing through the origin and perpendicular to $L_1$. If the locus of the point of intersection of $L_1$ and $L_2$ is …

Let $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,a>0,b>0$, 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 $\frac{\sqrt{11}}{2}$, then value of $a^2+b^2$ is equal to ____.

Let $\lambda x-2y=\mu$ be a tangent to the hyperbola $a^2{x}^2-y^2=b^2$. Then ${\left(\frac{\lambda }{a}\right)}^2-{\left(\frac{\mu }{b}\right)}^2$ is equal to:

Let the hyperbola $H:\frac{x^2}{a^2}-y^2=1$ and the ellipse $E:3x^2+4y^2=12$ be such that the length of latus rectum of H is equal to the length of latus rectum of E. If $e_H$ and $e_E$ are the eccentricities of H and E respectively, then the value of …

Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be $\frac{5}{4}$. If the equation of the normal at the point $\left(\frac{8}{\sqrt{5}},\frac{12}{5}\right)$ on the hyperbola is $8\sqrt{5}x+\beta y=\lambda$, then $\lambda -\beta$ is equal to

The locus of the mid points of the chords of the hyperbola $x^2-y^2=4$, which touch the parabola $y^2=8x$, is: