Conic Sections

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 :

If OB is the semi-minor axis of an ellipse, $F_1$ and $F_2$ are its foci and the angle between $F_1B$ and $F_2B$ is a right angle, then the square of the eccentricity of the ellipse is :

A stair-case of length $l$ rests against a vertical wall and a floor of a room,. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio 1 : 2. If the stair-case begins to slide on the floor, then the locus of P 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 :

Let $P(3\sec\theta,2\tan\theta)$ and $Q(3\sec\phi,2\tan\phi)$ where $\theta+\phi=\dfrac\pi2$, be two distinct points on the hyperbola $\dfrac{x^2}9-\dfrac{y^2}4=1$. Then the ordinate of the point of intersection of the normals at P and Q is :

Two tangents are drawn from a point $(-2,-1)$ to the curve, $y^2=4x$. If $\alpha$ is the angle between them, then $|\tan\alpha|$ is equal to :

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 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 :

The tangent at an extremity (in the first quadrant) of latus rectum of the hyperbola $\dfrac{x^2}4-\dfrac{y^2}5=1$, meets $x$ - axis and $y$ - axis at A and B respectively. Then $(OA)^2-(OB)^2$, where O is the origin, equals :

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 $\dfrac8{\sqrt5}\Delta_1-8\Delta_2=$

Length of chord $PQ$ is

If chord $PQ$ subtends an angle $\theta$ at the vertex of $y^2=4ax$, then $\tan\theta=$

A line $L:y=mx+3$ meets $y$-axis at $E(0,3)$ and the arc of the parabola $y^2=16x$, $0\le y\le6$ at the point $F(x_0,y_0)$. The tangent to the parabola at $F(x_0,y_0)$ intersects the $y$-axis at $G(0,y_1)$. The slope $m$ of the line $L$ is chosen such that the area of the triangle $EFG$ has a local maximum.

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

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

Tangents are drawn to the hyperbola $\dfrac{x^2}9-\dfrac{y^2}4=1$, parallel to the straight line $2x-y=1$. The points of contact of the tangents on the hyperbola are

Let $S$ be the focus of the parabola $y^2=8x$ and let $PQ$ be the common chord of the circle $x^2+y^2-2x-4y=0$ and the given parabola. The area of the triangle $PQS$ 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

Consider the parabola $y^2=8x$. Let $\Delta_1$ be the area of the triangle formed by the end points of its latus rectum and the point $P\left(\frac12,2\right)$ on the parabola, and $\Delta_2$ be the area of the triangle formed by drawing tangents at $P$ and at the end points of the latus rectum. Then $\dfrac{\Delta_1}{\Delta_2}$ is

Let $P(6,3)$ be a point on the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$. If the normal at the point $P$ intersects the $x$-axis at $(9,0)$, then the eccentricity of the hyperbola is