Conic Sections

Consider the parabola $y^2 = 4x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P = (-2, 1)$ meet the parabola at $P_1$ and $P_2$. Let $Q_1$ and $Q_2$ be points on the lines $SP_1$ and $SP_2$ respectively such that $PQ_1$ is perpendicular to $SP_1$ and $PQ_2$ is perpendicular to $SP_2$. Then, which of the following is/are TRUE?

Consider the ellipse
$$\frac{x^2}{4} + \frac{y^2}{3} = 1.$$
Let $H(\alpha, 0)$, $0 < \alpha < 2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\phi$ with the positive $x$-axis.

The correct option is:

Consider the hyperbola
$$\frac{x^2}{100} - \frac{y^2}{64} = 1$$
with foci at $S$ and $S_1$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle SPS_1 = \alpha$, with $\alpha < \dfrac\pi2$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S_1P$ at $P_1$. Let $\delta$ be the distance of $P$ from the straight line $SP_1$, and $\beta = S_1P$. Then the greatest integer less than or equal to $\dfrac{\beta\delta}{9}\sin\dfrac\alpha2$ is _____.

Let $\lambda x-2y=\mu$ be a tangent to the hyperbola $a^2x^2-y^2=b^2$. Then $\left(\dfrac\lambda a\right)^2-\left(\dfrac\mu b\right)^2$ is equal to :

Let $A\left(\dfrac3{\sqrt a},\sqrt a\right)$, $a>0$, be a fixed point in the $xy$-plane. The image of $A$ in $y$-axis be $B$ and the image of $B$ in $x$-axis be $C$. If $D(3\cos\theta,a\sin\theta)$ is a point in the fourth quadrant such that the maximum area of $\Delta ACD$ is 12 square units, then $a$ is equal to ______.

If two tangents drawn from a point $(\alpha,\beta)$ lying on the ellipse $25x^2+4y^2=1$ to the parabola $y^2=4x$ are such that the slope of one tangent is four times the other, then the value of $(10\alpha+5)^2+(16\beta^2+50)^2$ equals ______.

A particle is moving in the $xy$-plane along a curve $C$ passing through the point $(3,3)$. The tangent to the curve $C$ at the point $P$ meets the $x$-axis at $Q$. If the $y$-axis bisects the segment $PQ$, then $C$ is a parabola with

Let the maximum area of the triangle that can be inscribed in the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}4=1$, $a>2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the $y$-axis, be $6\sqrt3$. Then the eccentricity of the ellipse is:

Let the hyperbola $H:\dfrac{x^2}{a^2}-y^2=1$ and the ellipse $E:3x^2+4y^2=12$ be such that the length of latus rectum of $H$ is equal to the length of latus rectum of $E$. If $e_H$ and $e_E$ are the eccentricities of $H$ and $E$ respectively, then the value of $12\left(e_H^2+e_E^2\right)$ is equal to ______.

Let $\mathrm P_1$ be a parabola with vertex $(3,2)$ and focus $(4,4)$ and $\mathrm P_2$ be its mirror image with respect to the line $x+2y=6$. Then the directrix of $\mathrm P_2$ is $x+2y=$ ______.

If $y=m_1x+c_1$ and $y=m_2x+c_2$, $m_1\ne m_2$ are two common tangents of circle $x^2+y^2=2$ and parabola $y^2=x$, then the value of $8|m_1m_2|$ is equal to :

Let $x=2t$, $y=\dfrac{t^2}3$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $SA\perp BA$, where A is any point on the conic. If k is the ordinate of the centroid of the $\Delta SAB$, then $\displaystyle\lim_{t\to1}k$ is equal to :

The area of the region enclosed between the parabolas $y^2=2x-1$ and $y^2=4x-3$ is

If the line $y=4+kx$, $k>0$, is the tangent to the parabola $y=x-x^2$ at the point $P$ and $V$ is the vertex of the parabola, then the slope of the line through $P$ and $V$ is:

The line $y=x+1$ meets the ellipse $\dfrac{x^2}4+\dfrac{y^2}2=1$ at two points $P$ and $Q$. If $r$ is the radius of the circle with $PQ$ as diameter then $(3r)^2$ is equal to :

Let the eccentricity of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ be $\dfrac54$. If the equation of the normal at the point $\left(\dfrac8{\sqrt5},\dfrac{12}5\right)$ on the hyperbola is $8\sqrt5x+\beta y=\lambda$, then $\lambda-\beta$ is equal to ______.

Let the normal at the point P on the parabola $y^2=6x$ pass through the point $(5,-8)$. If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is :

Let the common tangents to the curves $4\left(x^2+y^2\right)=9$ and $y^2=4x$ intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\dfrac l{e^2}$ is equal to ______.

The area of the region bounded by $y^2=8x$ and $y^2=16(3-x)$ is equal to:

If $m$ is the slope of a common tangent to the curves $\dfrac{x^2}{16}+\dfrac{y^2}9=1$ and $x^2+y^2=12$, then $12m^2$ is equal to :