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

Consider an ellipse $E$,a hyperbola $H$,and a parabola $P$ such that each curve has the focus at $(2, 3)$ and the corresponding directrix is $x + y - 10 = 0$. If $(\alpha, \alpha_1)$,$(\beta, \beta_1)$,and $(\gamma, \gamma_1)$ are the nearest vertices of the ellipse,hyperbola,and parabola to the given directrix respectively,then:

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$,respectively,and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$,respectively. Then which of the following statements is(are) true?
$(A)$ The area of the quadrilateral $A_1 A_2 A_3 A_4$ is $35$ square units.
$(B)$ The area of the quadrilateral $A_1 A_2 A_3 A_4$ is $36$ square units.
$(C)$ The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-3,0)$.
$(D)$ The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-6,0)$.

An equation for the line that passes through $(10, -1)$ and is perpendicular to $y = \frac{x^2}{4} - 2$ is

$A$ bar of length $20$ units moves with its ends on two fixed straight lines at right angles. $A$ point $P$ marked on the bar at a distance of $8$ units from one end describes a conic whose eccentricity is

An ellipse has eccentricity $\frac{1}{2}$ and one focus at the point $P\left( \frac{1}{2}, 1 \right)$. Its one directrix is the common tangent nearer to the point $P$, to the circle $x^2 + y^2 = 1$ and the hyperbola $x^2 - y^2 = 1$. The equation of the ellipse in the standard form is: