Conic Sections

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 one root of the quadratic equation in $x$:
$$(k^2-15k+27)x^2+9(k-1)x+18=0$$
be twice the other. Then the length of the latus rectum of the parabola $y^2=6kx$ 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:

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:

Let chord PQ of length $3\sqrt{13}$ of the parabola $y^2=12x$ be such that the ordinates of points P and Q are in the ratio 1:2. If the chord PQ subtends an angle $\alpha$ at the focus of the parabola, then $\sin\alpha$ is equal to:

The area of the region $\{(x,y):0\le y\le6-x,\ y^2\ge4x-3,\ x\ge0\}$ 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 $x=9$ 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 O be the vertex of the parabola $y^2=4x$ and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum 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 $S$ 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

Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2 = x$, such that the area of the region enclosed between the parabola and the chord is $\dfrac43$. Let $\mathcal{R}$ denote the region lying in the first quadrant, enclosed by the parabola $y^2 = x$, the curve $S$, and the lines $x = 1$ and $x = 4$.

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

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 $S = \left\{(x,y) \in \mathbb{R}\times\mathbb{R} : x \ge 0,\ y \ge 0,\ y^2 \le 4x,\ y^2 \le 12 - 2x \text{ and } 3y + \sqrt8\,x \le 5\sqrt8\right\}$. If the area of the region $S$ is $\alpha\sqrt2$, then $\alpha$ is equal to

Let $A_1, B_1, C_1$ be three points in the $xy$-plane. Suppose that the lines $A_1C_1$ and $B_1C_1$ are tangents to the curve $y^2 = 8x$ at $A_1$ and $B_1$, respectively. If $O = (0,0)$ and $C_1 = (-4, 0)$, then which of the following statements is (are) TRUE?

A normal with slope $\dfrac{1}{\sqrt6}$ is drawn from the point $(0, -\alpha)$ to the parabola $x^2 = -4ay$, where $a > 0$. Let $L$ be the line passing through $(0, -\alpha)$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B$. Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment $AB$. If $r : s = 1 : 16$, then the value of $24a$ is _____.

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?

Let $P$ be a point on the parabola $y^2 = 4ax$, where $a > 0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $PFQ$, where $F$ is the focus of the parabola, is 120. If the slope $m$ of the normal and $a$ are both positive integers, then the pair $(a, m)$ is