Hyperbola

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Consider the ellipse $E$ given by $\dfrac{x^2}{18}+\dfrac{y^2}{12}=1$. Let $H$ be the hyperbola whose eccentricity is the reciprocal of the eccentricity of $E$ and whose foci are the same as that of $E$. Let $P$ and $Q$ be the points of intersection of $H$ and the parabola $\sqrt5\,y=x^2$ in the first quadrant. Let $d$ be the distance between $P$ and $Q$.

If $a$ and $b$ are the integers such that $d^2=a+b\sqrt5$, then the value of $a-b$ is ________.

Let O be the origin, and P and Q be two points on the rectangular hyperbola $xy=12$ such that the mid point of the line segment PQ is $\left(\dfrac12,-\dfrac12\right)$. Then the area of the triangle OPQ equals :

If the area of the region bounded by $16x^2-9y^2=144$ and $8x-3y=24$ is A, then $3\left(A+6\log_e(3)\right)$ is equal to ________.

Let $H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is $\dfrac83$. If the line $x=\alpha$ intersects the hyperbola H at the points A and B such that the area of the triangle AOB is $4\sqrt{15}$, where O is the origin, then $\alpha^2$ equals

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

Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2-ax+2=0$. Let the sets
$\{a\in\mathbb R:e_1\text{ and }e_2\text{ are the eccentricities of hyperbolas}\}=(\alpha,\beta)$, and
$\{a\in\mathbb R:e_1\text{ and }e_2\text{ are the eccentricities of an ellipse and a hyperbola, respectively}\}=(\gamma,\infty)$.
Then $\alpha^2+\beta^2+\gamma^2$ is equal to:

If the eccentricity $e$ of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$, passing through $\left(6,4\sqrt3\right)$, satisfies $15(e^2+1)=34e$, then the length of the latus rectum of the hyperbola $\dfrac{x^2}{b^2}-\dfrac{y^2}{2(a^2+1)}=1$ is:

The eccentricity of an ellipse E with centre at the origin O is $\dfrac{\sqrt3}2$ and its directrices are $x=\pm\dfrac{4\sqrt6}3$. Let $H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ be a hyperbola whose eccentricity is equal to the length of semi-major axis of E, and whose length of latus rectum is equal to the length of minor axis of E. Then the distance between the foci of H is :

Let $S$ denote the locus of the point of intersection of the pair of lines

$$4x - 3y = 12\alpha,\qquad 4\alpha x + 3\alpha y = 12,$$

where $\alpha$ varies over the set of non-zero real numbers. Let $T$ be the tangent to $S$ passing through the points $(p, 0)$ and $(0, q)$, $q > 0$, and parallel to the line $4x - \dfrac{3}{\sqrt2}y = 0$.

Then the value of $pq$ is

Consider the hyperbola
$$\frac{x^2}{100} - \frac{y^2}{64} = 1$$
with foci at $S$ and $S_1$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle SPS_1 = \alpha$, with $\alpha < \dfrac\pi2$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S_1P$ at $P_1$. Let $\delta$ be the distance of $P$ from the straight line $SP_1$, and $\beta = S_1P$. Then the greatest integer less than or equal to $\dfrac{\beta\delta}{9}\sin\dfrac\alpha2$ is _____.

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

Let the hyperbola $H:\dfrac{x^2}{a^2}-y^2=1$ 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 $12\left(e_H^2+e_E^2\right)$ is equal to ______.

Let the eccentricity of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ be $\dfrac54$. If the equation of the normal at the point $\left(\dfrac8{\sqrt5},\dfrac{12}5\right)$ on the hyperbola is $8\sqrt5x+\beta y=\lambda$, then $\lambda-\beta$ is equal to ______.

The normal to the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}9=1$ at the point $\left(8,3\sqrt3\right)$ on it passes through the point :

Let the eccentricity of the hyperbola $H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ be $\sqrt{\dfrac52}$ and length of its latus rectum be $6\sqrt2$. If $y=2x+c$ is a tangent to the hyperbola H, then the value of $c^2$ is equal to

Let $a>0$, $b>0$. Let $e$ and $l$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$. Let $e'$ and $l'$ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If $e^2=\dfrac{11}{14}l$ and $\left(e'\right)^2=\dfrac{11}8l'$, then the value of $77a+44b$ is equal to :

Let $H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$, $a>0$, $b>0$, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4\left(2\sqrt2+\sqrt{14}\right)$. If the eccentricity H is $\dfrac{\sqrt{11}}2$, then the value of $a^2+b^2$ is equal to ______.

Let the eccentricity of the ellipse $x^2+a^2y^2=25a^2$ be b times the eccentricity of the hyperbola $x^2-a^2y^2=5$, where a is the minimum distance between the curves $y=e^x$ and $y=\log_ex$. Then $a^2+\dfrac1{b^2}$ is equal to :

Let the foci of the ellipse $\dfrac{x^2}{16}+\dfrac{y^2}7=1$ and the hyperbola $\dfrac{x^2}{144}-\dfrac{y^2}\alpha=\dfrac1{25}$ coincide. Then the length of the latus rectum of the hyperbola is :