Parabola

Let $T$ be the tangent to the parabola $y^2=16x$ at the point $(64,32)$. Let $L$ be the tangent to the same parabola at another point $(x_1,y_1)$ on the parabola. If $L$ and $T$ are perpendicular to each other, then the distance between the point $(x_1,y_1)$ and the focus of the parabola, is

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 the parabola $y=x^2+px+q$ passing through the point $(1,-1)$ be such that the distance between its vertex and the $x$-axis is minimum. Then the value of $p^2+q^2$ is :

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 ________.

The area of the region bounded by the curves $x+3y^2=0$ and $x+4y^2=1$ is equal to:

Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola $y^2=16x$. Let the vertex B containing the right angle be $(4,8)$ and the locus of the centroid of $\Delta ABC$ be a conic $C_0$. Then three times the length of latus rectum of $C_0$ is ________.

Let P be a moving point on the circle $x^2+y^2-6x-8y+21=0$. Then, the maximum distance of P from the vertex of the parabola $x^2+6x+y+13=0$ is equal to:

Let the point P be the vertex of the parabola $y=x^2-6x+12$. If a line passing through the point P intersects the circle $x^2+y^2-2x-4y+3=0$ at the points R and S, then the maximum value of $(PR+PS)^2$ is :

Let the directrix of the parabola $P:y^2=8x$, cut $x$-axis at the point A. Let $B(\alpha,\beta)$, $\alpha>1$, be a point on P such that the slope of AB is $3/5$. If BC is a focal chord of P, then six times the area of $\Delta ABC$ 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 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:

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 $S$ 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 $S = \left\{(x,y) \in \mathbb{R}\times\mathbb{R} : x \ge 0,\ y \ge 0,\ y^2 \le 4x,\ y^2 \le 12 - 2x \text{ and } 3y + \sqrt8\,x \le 5\sqrt8\right\}$. 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

Consider the parabola $y^2 = 4x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P = (-2, 1)$ meet the parabola at $P_1$ and $P_2$. Let $Q_1$ and $Q_2$ be points on the lines $SP_1$ and $SP_2$ respectively such that $PQ_1$ is perpendicular to $SP_1$ and $PQ_2$ is perpendicular to $SP_2$. Then, which of the following is/are TRUE?