Ellipse

The minimum area of a triangle formed by any tangent to the ellipse $\dfrac{x^2}{16}+\dfrac{y^2}{81}=1$ and the co-ordinate axes is :

A vertical line passing through the point $(h,0)$ intersects the ellipse $\dfrac{x^2}4+\dfrac{y^2}3=1$ at the points $P$ and $Q$. Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$. If $\Delta(h)=$ area of the triangle $PQR$, $\Delta_1=\displaystyle\max_{1/2\le h\le1}\Delta(h)$ and $\Delta_2=\displaystyle\min_{1/2\le h\le1}\Delta(h)$, then $\dfrac8{\sqrt5}\Delta_1-8\Delta_2=$

The ellipse $E_1:\dfrac{x^2}9+\dfrac{y^2}4=1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_2$ passing through the point $(0,4)$ circumscribes the rectangle $R$. The eccentricity of the ellipse $E_2$ is

Let the eccentricity of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ be reciprocal to that of the ellipse $x^2+4y^2=4$. If the hyperbola passes through a focus of the ellipse, then

The coordinates of A and B are

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

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 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