Conic Sections

The locus of the mid point of the line segment joining the point $(4,3)$ and the points on the ellipse $x^2+2y^2=4$ is an ellipse with eccentricity :

The normal to the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}9=1$ at the point $\left(8,3\sqrt3\right)$ on it passes through the point :

Let a line $L_1$ be tangent to the hyperbola $\dfrac{x^2}{16}-\dfrac{y^2}4=1$ and let $L_2$ be the line passing through the origin and perpendicular to $L_1$. If the locus of the point of intersection of $L_1$ and $L_2$ is $\left(x^2+y^2\right)^2=\alpha x^2+\beta y^2$, then $\alpha+\beta$ is equal to ______.

Let the eccentricity of an ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, $a>b$, be $\dfrac14$. If this ellipse passes through the point $\left(-4\sqrt{\dfrac25},3\right)$, then $a^2+b^2$ is equal to :

A circle of radius 2 unit passes through the vertex and the focus of the parabola $y^2=2x$ and touches the parabola $y=\left(x-\dfrac14\right)^2+\alpha$, where $\alpha>0$. Then $(4\alpha-8)^2$ is equal to ______.

The number of points of intersection of $|z-(4+3i)|=2$ and $|z|+|z-4|=6$, $z\in\mathbb C$, is

If the equation of the parabola, whose vertex is at $(5,4)$ and the directrix is $3x+y-29=0$, is $x^2+ay^2+bxy+cx+dy+k=0$, then $a+b+c+d+k$ is equal to

Let the eccentricity of the hyperbola $H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ be $\sqrt{\dfrac52}$ and length of its latus rectum be $6\sqrt2$. If $y=2x+c$ is a tangent to the hyperbola H, then the value of $c^2$ is equal to

Let $a>0$, $b>0$. Let $e$ and $l$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$. Let $e'$ and $l'$ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If $e^2=\dfrac{11}{14}l$ and $\left(e'\right)^2=\dfrac{11}8l'$, then the value of $77a+44b$ is equal to :

If vertex of a parabola is $(2,-1)$ and the equation of its directrix is $4x-3y=21$, then the length of its latus rectum is :

Let PQ be a focal chord of the parabola $y^2=4x$ such that it subtends an angle of $\dfrac\pi2$ at the point $(3,0)$. Let the line segment PQ be also a focal chord of the ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, $a^2>b^2$. If $e$ is the eccentricity of the ellipse E, then the value of $\dfrac1{e^2}$ is equal to :

Let $H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$, $a>0$, $b>0$, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4\left(2\sqrt2+\sqrt{14}\right)$. If the eccentricity H is $\dfrac{\sqrt{11}}2$, then the value of $a^2+b^2$ is equal to ______.

Let $P:y^2=4ax$, $a>0$ be a parabola with focus S. Let the tangents to the parabola P make an angle of $\dfrac\pi4$ with the line $y=3x+5$ touch the parabola P at A and B. Then the value of $a$ for which A, B and S are collinear is

For real numbers $a,b$ $(a>b>0)$, let
$$\text{Area}\left\{(x,y):x^2+y^2\le a^2\text{ and }\frac{x^2}{a^2}+\frac{y^2}{b^2}\ge1\right\}=30\pi$$
and
$$\text{Area}\left\{(x,y):x^2+y^2\ge b^2\text{ and }\frac{x^2}{a^2}+\frac{y^2}{b^2}\le1\right\}=18\pi$$
Then the value of $(a-b)^2$ is equal to ______.

Let the eccentricity of the ellipse $x^2+a^2y^2=25a^2$ be b times the eccentricity of the hyperbola $x^2-a^2y^2=5$, where a is the minimum distance between the curves $y=e^x$ and $y=\log_ex$. Then $a^2+\dfrac1{b^2}$ is equal to :

Let PQ be a focal chord of length 6.25 units of the parabola $y^2=4x$. If O is the vertex of the parabola, then 10 times the area (in sq. units) of $\Delta$POQ is equal to ______.

Let the equation of two diameters of a circle $x^2+y^2-2x+2fy+1=0$ be $2px-y=1$ and $2x+py=4p$. Then the slope $m\in(0,\infty)$ of the tangent to the hyperbola $3x^2-y^2=3$ passing through the centre of the circle is equal to ______.

The sum of diameters of the circles that touch (i) the parabola $75x^2=64(5y-3)$ at the point $\left(\dfrac85,\dfrac65\right)$ and (ii) the $y$-axis, is equal to ______.

If the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ meets the line $\dfrac x7+\dfrac y{2\sqrt6}=1$ on the $x$-axis and the line $\dfrac x7-\dfrac y{2\sqrt6}=1$ on the $y$-axis, then the eccentricity of the ellipse is

The tangents at the points $A(1,3)$ and $B(1,-1)$ on the parabola $y^2-2x-2y=1$ meet at the point P. Then the area (in unit$^2$) of the triangle PAB is: