Ellipse

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 ______.

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 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 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 :

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 :

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

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 :

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

Let the foci of the ellipse $\dfrac{x^2}{16}+\dfrac{y^2}7=1$ and the hyperbola $\dfrac{x^2}{144}-\dfrac{y^2}\alpha=\dfrac1{25}$ coincide. Then the length of the latus rectum of the hyperbola is :

The acute angle between the pair of tangents drawn to the ellipse $2x^2+3y^2=5$ from the point $(1,3)$ is

An ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ passes through the vertices of the hyperbola $H:\dfrac{x^2}{49}-\dfrac{y^2}{64}=-1$. Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be $\dfrac12$. If $l$ is the length of the latus rectum of the ellipse E, then the value of $113l$ is equal to ______.

If the length of the latus rectum of the ellipse $x^2+4y^2+2x+8y-\lambda=0$ is 4, and $l$ is the length of its major axis, then $\lambda+l$ is equal to ______.

A common tangent T to the curves $C_1:\dfrac{x^2}4+\dfrac{y^2}9=1$ and $C_2:\dfrac{x^2}{42}-\dfrac{y^2}{143}=1$ does not pass through the fourth quadrant. If T touches $C_1$ at $(x_1,y_1)$ and $C_2$ at $(x_2,y_2)$, then $|2x_1+x_2|$ is equal to ______.

For the hyperbola $H:x^2-y^2=1$ and the ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, $a>b>0$, let the

(1) eccentricity of E be reciprocal of the eccentricity of H, and

(2) the line $y=\sqrt{\dfrac52}x+K$ be a common tangent of E and H.

Then $4\left(a^2+b^2\right)$ is equal to ______.

Let the tangents at the points P and Q on the ellipse $\dfrac{x^2}2+\dfrac{y^2}4=1$ meet at the point $R\left(\sqrt2,2\sqrt2-2\right)$. If S is the focus of the ellipse on its negative major axis, then $SP^2+SQ^2$ is equal to ______.

Let a line L pass through the point of intersection of the lines $bx+10y-8=0$ and $2x-3y=0$, $b\in\mathbf R-\left\{\frac43\right\}$. If the line L also passes through the point (1, 1) and touches the circle $17(x^2+y^2)=16$, then the eccentricity of the ellipse $\dfrac{x^2}5+\dfrac{y^2}{b^2}=1$ is :

Let $S=\left\{(x,y)\in\mathbb N\times\mathbb N:9(x-3)^2+16(y-4)^2\le144\right\}$ and $T=\left\{(x,y)\in\mathbb R\times\mathbb R:(x-7)^2+(y-4)^2\le36\right\}$. Then $n(S\cap T)$ is equal to _______.

If foci of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{12}=1$ and the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}9=1$ coincide, then the eccentricity of the hyperbola, with foci $(0,\pm\sqrt{10})$ passing through the point (a, a + 1), is