Conic Sections

P and Q are two distinct points on the parabola, $y^2=4x$, with parameters t and $t_1$ respectively. If the normal at P passes through Q, then the minimum value of $t_1^2$ is :

A hyperbola whose transverse axis is along the major axis of the conic, $\dfrac{x^2}3+\dfrac{y^2}4=4$ and has vertices at the foci of this conic. If the eccentricity of the hyperbola is $\dfrac32$, then which of the following points does NOT lie on it ?

Let the curve $C$ be the mirror image of the parabola $y^2=4x$ with respect to the line $x+y+4=0$. If $A$ and $B$ are the points of intersection of $C$ with the line $y=-5$, then the distance between $A$ and $B$ is

If the normals of the parabola $y^2=4x$ drawn at the end points of its latus rectum are tangents to the circle $(x-3)^2+(y+2)^2=r^2$, then the value of $r^2$ is

Let $P$ and $Q$ be distinct points on the parabola $y^2=2x$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\Delta OPQ$ is $3\sqrt2$, then which of the following is (are) the coordinates of $P$?

Suppose that the foci of the ellipse $\dfrac{x^2}9+\dfrac{y^2}5=1$ are $(f_1,0)$ and $(f_2,0)$ where $f_1>0$ and $f_2<0$. Let $P_1$ and $P_2$ be two parabolas with a common vertex at $(0,0)$ and with foci at $(f_1,0)$ and $(2f_2,0)$, respectively. Let $T_1$ be a tangent to $P_1$ which passes through $(2f_2,0)$ and $T_2$ be a tangent to $P_2$ which passes through $(f_1,0)$. If $m_1$ is the slope of $T_1$ and $m_2$ is the slope of $T_2$, then the value of $\left(\dfrac1{m_1^2}+m_2^2\right)$ is

Let $E_1$ and $E_2$ be two ellipses whose centers are at the origin. The major axes of $E_1$ and $E_2$ lie along the $x$-axis and the $y$-axis, respectively. Let $S$ be the circle $x^2+(y-1)^2=2$. The straight line $x+y=3$ touches the curves $S$, $E_1$ and $E_2$ at $P$, $Q$ and $R$, respectively. Suppose that $PQ=PR=\dfrac{2\sqrt2}3$. If $e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2$, respectively, then the correct expression(s) is(are)

Consider the hyperbola $H:x^2-y^2=1$ and a circle $S$ with center $N(x_2,0)$. Suppose that $H$ and $S$ touch each other at a point $P(x_1,y_1)$ with $x_1>1$ and $y_1>0$. The common tangent to $H$ and $S$ at $P$ intersects the $x$-axis at point $M$. If $(l,m)$ is the centroid of the triangle $\Delta PMN$, then the correct expression(s) is(are)

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse $\dfrac{x^2}9+\dfrac{y^2}5=1$, is :

Let O be the vertex and Q be any point on the parabola, $x^2=8y$. If the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is :

If the tangent to the conic, $y-6=x^2$ at (2, 10) touches the circle, $x^2+y^2+8x-2y=k$ (for some fixed k) at a point (α, β) ; then (α, β) is :

An ellipse passes through the foci of the hyperbola, $9x^2-4y^2=36$ and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is $\dfrac12$, then which of the following points does not lie on the ellipse ?

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is :

Let PQ be a double ordinate of the parabola, $y^2=-4x$, where P lies in the second quadrant. If R divides PQ in the ratio 2 : 1, then the locus of R is :

The common tangents to the circle $x^2+y^2=2$ and the parabola $y^2=8x$ touch the circle at the points $P$, $Q$ and the parabola at the points $R$, $S$. Then the area of the quadrilateral $PQRS$ is

The value of $r$ is

If $st=1$, then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

Match List I with List II and select the correct answer using the code given below the lists.

The area of the region described by $A=\{(x,y):x^2+y^2\le1\text{ and }y^2\le1-x\}$ is :

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^2+3y^2=6$ on any tangent to it is :