Parabola

The orthocentre of the triangle $F_1MN$ is

If the tangents to the ellipse at $M$ and $N$ meet at $R$ and the normal to the parabola at $M$ meets the $x$-axis at $Q$, then the ratio of area of the triangle $MQR$ to area of the quadrilateral $MF_1NF_2$ is

The centres of those circles which touch the circle, $x^2+y^2-8x-8y-4=0$, externally and also touch the $x$-axis, lie on :

Let P be the point on the parabola, $y^2=8x$ which is at a minimum distance from the centre C of the circle, $x^2+(y+6)^2=1$. Then the equation of the circle, passing through C and having its centre at P is :

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 :

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

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 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 :

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

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

The slope of the line touching both the parabolas $y^2=4x$ and $x^2=-32y$ is :

Let $A=\left\{(x,y):y^2\le4x,\ y-2x\ge-4\right\}$. The area (in square units) of the region A is :

Let $L_1$ be the length of the common chord of the curves $x^2+y^2=9$ and $y^2=8x$, and $L_2$ be the length of the latus rectum of $y^2=8x$, then :

A chord is drawn through the focus of the parabola $y^2=6x$ such that its distance from the vertex of this parabola is $\dfrac{\sqrt5}2$, then its slope can be :

Length of chord $PQ$ is