Conic Sections

Let $P$ be the point on the parabola $y=x^2$ such that the slope of the tangent to the parabola at the point $P$ is $4$. Let $Q$ be the point in the first quadrant lying on the circle $x^2+y^2=2$ such that the slope of the tangent to the circle at the point $Q$ is $-1$. Let $R$ be the point in the first quadrant lying on the ellipse $x^2+4y^2=8$ such that the slope of the tangent to the ellipse at the point $R$ is $-\frac12$. Then the radius of the circle passing through the points $P,Q$ and $R$ is

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

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 $P$ be the point in the first quadrant where the given ellipses intersect. If $\theta$ is the acute angle between the tangents to the given ellipses at the point $P$, then the value of $4\tan\theta$ 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 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 :

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 :

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 ________.

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 ________.

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 $H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ 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

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 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 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 :