Ellipse

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 ________.

If $\alpha$ is the area of the common region that lies inside both the given ellipses, then the value of $\cot\alpha$ is ________.

Let an ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, $a<b$, pass through the point $(4,3)$ and have eccentricity $\dfrac{\sqrt5}3$. Then the length of its latus rectum is :

Let A be the point $(3,0)$ and circles with variable diameter AB touch the circle $x^2+y^2=36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is $e$, then $72e^2$ is equal to ________.

Consider the parabola $P:y^2=4kx$ and the ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$. Let the line segment joining the points of intersection of P and E, be their latus rectums. If the eccentricity of E is $e$, then $e^2+2\sqrt2$ is equal to ________.

Let $P(3\cos\alpha,2\sin\alpha)$, $\alpha\ne0$, be a point on the ellipse $\dfrac{x^2}9+\dfrac{y^2}4=1$, Q be a point on the circle $x^2+y^2-14x-14y+82=0$ and R be a point on the line $x+y=5$ such that the centroid of the triangle PQR is $\left(2+\cos\alpha,\ 3+\dfrac23\sin\alpha\right)$. Then the sum of the ordinates of all possible points R is:

Let a focus of the ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ be $S(4,0)$ and its eccentricity be $\dfrac45$. If the point $P(3,\alpha)$ lies on E and O is the origin, then the area of $\Delta POS$ is equal to:

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:

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 $x=9$ be a directrix of an ellipse E, whose centre is at the origin and eccentricity is $\dfrac13$. Let $P(\alpha,0)$, $\alpha>0$, be a focus of E and AB be a chord passing through P. Then the locus of the mid point of AB is :

The number of values of $z\in\mathbb C$, satisfying the equations $|z-(4+8i)|=\sqrt{10}$ and $|z-(3+5i)|+|z-(5+11i)|=4\sqrt5$, is :

Let $\dfrac{x^2}{f\left(a^2+7a+3\right)}+\dfrac{y^2}{f(3a+15)}=1$ represent an ellipse with major axis along $y$-axis, where $f$ is a strictly decreasing positive function on $\mathbf R$. If the set of all possible values of a is $\mathbf R-[\alpha,\beta]$, then $\alpha^2+\beta^2$ is equal to :

Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ be two distinct points on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ such that $y_1 > 0$, and $y_2 > 0$. Let $C$ denote the circle $x^2 + y^2 = 9$, and $M$ be the point $(3, 0)$.

Suppose the line $x = x_1$ intersects $C$ at $R$, and the line $x = x_2$ intersects $C$ at $S$, such that the $y$-coordinates of $R$ and $S$ are positive. Let $\angle ROM = \dfrac{\pi}{6}$ and $\angle SOM = \dfrac{\pi}{3}$, where $O$ denotes the origin $(0, 0)$. Let $|XY|$ denote the length of the line segment $XY$.

Then which of the following statements is (are) TRUE?

Consider the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$. Let $S(p,q)$ be a point in the first quadrant such that $\dfrac{p^2}{9} + \dfrac{q^2}{4} > 1$. Two tangents are drawn from $S$ to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point $T$ in the fourth quadrant. Let $R$ be the vertex of the ellipse with positive $x$-coordinate and $O$ be the center of the ellipse. If the area of the triangle $\Delta ORT$ is $\dfrac{3}{2}$, then which of the following options is correct?

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E : \dfrac{x^2}{6} + \dfrac{y^2}{3} = 1$ and the parabola $P : y^2 = 12x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?

Consider the ellipse
$$\frac{x^2}{4} + \frac{y^2}{3} = 1.$$
Let $H(\alpha, 0)$, $0 < \alpha < 2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\phi$ with the positive $x$-axis.

The correct option is:

Let $A\left(\dfrac3{\sqrt a},\sqrt a\right)$, $a>0$, be a fixed point in the $xy$-plane. The image of $A$ in $y$-axis be $B$ and the image of $B$ in $x$-axis be $C$. If $D(3\cos\theta,a\sin\theta)$ is a point in the fourth quadrant such that the maximum area of $\Delta ACD$ is 12 square units, then $a$ is equal to ______.

If two tangents drawn from a point $(\alpha,\beta)$ lying on the ellipse $25x^2+4y^2=1$ to the parabola $y^2=4x$ are such that the slope of one tangent is four times the other, then the value of $(10\alpha+5)^2+(16\beta^2+50)^2$ equals ______.

Let the maximum area of the triangle that can be inscribed in the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}4=1$, $a>2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the $y$-axis, be $6\sqrt3$. Then the eccentricity of the ellipse is: