Conic Sections

Let $(x,y)$ be any point on the parabola $y^2=4x$. Let $P$ be the point that divides the line segment from $(0,0)$ to $(x,y)$ in the ratio $1:3$. Then the locus of $P$ is

Let $L$ be a normal to the parabola $y^2=4x$. If $L$ passes through the point $(9,6)$, then $L$ is given by

Let A and B be two distinct points on the parabola $y^2=4x$. If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be

Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

Equation of the circle with AB as its diameter is

The line $2x+y=1$ is tangent to the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$. If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is

The coordinates of A and B are

The equation of the locus of the point whose distances from the point P and the line AB are equal, is

Match the statements in Column-I with those in Column-II.

[Note: Here $z$ takes values in the complex plane and Im $z$ and Re $z$ denote, respectively, the imaginary part and the real part of $z$.]

The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse
$$x^2+9y^2=9$$
meets its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A$, $M$ and the origin $O$ is

In a triangle $ABC$ with fixed base $BC$, the vertex $A$ moves such that
$$\cos B+\cos C=4\sin^2\frac A2.$$
If $a$, $b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A$, $B$ and $C$, respectively, then

Match the conics in Column I with the statements/expressions in Column II.

The normal at a point $P$ on the ellipse $x^2+4y^2=16$ meets the $x$-axis at $Q$. If $M$ is the mid point of the line segment $PQ$, then the locus of $M$ intersects the latus rectums of the given ellipse at the points

An ellipse intersects the hyperbola $2x^2-2y^2=1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then

The tangent $PT$ and the normal $PN$ to the parabola $y^2=4ax$ at a point $P$ on it meet its axis at points $T$ and $N$, respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose

Consider the two curves
$$C_1:y^2=4x,\qquad C_2:x^2+y^2-6x+1=0.$$
Then,

Let $P(x_1,y_1)$ and $Q(x_2,y_2)$, $y_1<0$, $y_2<0$, be the end points of the latus rectum of the ellipse $x^2+4y^2=4$. The equations of parabolas with latus rectum $PQ$ are

Consider a branch of the hyperbola
$$x^2-2y^2-2\sqrt2x-4\sqrt2y-6=0$$
with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$, then the area of the triangle $ABC$ is

Consider three points $P=(-\sin(\beta-\alpha),-\cos\beta)$, $Q=(\cos(\beta-\alpha),\sin\beta)$ and $R=(\cos(\beta-\alpha+\theta),\sin(\beta-\theta))$, where $0<\alpha,\beta,\theta<\frac\pi4$. Then,

A hyperbola, having the transverse axis of length $2\sin\theta$, is confocal with the ellipse $3x^2+4y^2=12$. Then its equation is