If the line $x - 1 = 0$, is a directrix of the hyperbola $k x ^ { 2 } - y ^ { 2 } = 6$, then the hyperbola passes through the point
(1) $\left( - 2 \sqrt { 5 } , 6 \right)$
(2) $\left( - \sqrt { 5 } , 3 \right)$
(3) $\left( \sqrt { 5 } , - 2 \right)$
(4) $\left( 2 \sqrt { 5 } , 3 \sqrt { 6 } \right)$

If the equation of the parabola, whose vertex is at $( 5,4 )$ and the directrix is $3 x + y - 29 = 0$, is $x ^ { 2 } + a y ^ { 2 } + b x y + c x + d y + k = 0$, then $a + b + c + d + k$ is equal to
(1) 575
(2) - 575
(3) 576
(4) - 576

An ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ passes through the vertices of the hyperbola $H : \frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 64 } = - 1$. Let 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 $113 l$ is equal to $\_\_\_\_$ .

Let $P$ be a parabola with vertex $(2, 3)$ and directrix $2x + y = 6$. Let an ellipse $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $a > b$ of eccentricity $\dfrac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then the square of the length of the latus rectum of $E$, is
(1) $\dfrac{385}{8}$
(2) $\dfrac{347}{8}$
(3) $\dfrac{512}{25}$
(4) $\dfrac{656}{25}$

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E : \frac { ( x - 1 ) ^ { 2 } } { 100 } + \frac { ( y - 1 ) ^ { 2 } } { 75 } = 1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3 \alpha ^ { 2 } + 2 \beta ^ { 2 }$ is equal to
(1) 237
(2) 242
(3) 205
(4) 225

If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2}, 3\right)$ and the directrix $x + 2y = 0$ is $\alpha x^{2} + \beta y^{2} - \gamma xy - 30x - 60y + 225 = 0$, then $\alpha + \beta + \gamma$ is equal to:
(1) 7
(2) 9
(3) 8
(4) 6

On the coordinate plane, the equation of ellipse $\Gamma$ is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{6^{2}} = 1$ (where $a$ is a positive real number). If $\Gamma$ is scaled by a factor of 2 in the $x$-axis direction and by a factor of 3 in the $y$-axis direction with the origin $O$ as the center, the resulting new figure passes through the point $(18, 0)$. Which of the following options is a focus of $\Gamma$?
(1) $(0, 3\sqrt{3})$
(2) $(-3\sqrt{5}, 0)$
(3) $(0, 6\sqrt{13})$
(4) $(-3\sqrt{13}, 0)$
(5) $(9, 0)$

In the Cartesian coordinate plane, an ellipse with center at the origin and foci at points E and F is given below. The vertical line drawn from point F intersects the ellipse at points, and the point with positive y-coordinate is denoted by K. The equation of the line passing through points K and E is $\mathrm { y } = \mathrm { x } + 1$.
Accordingly, what is the value of a?
A) $\sqrt { 2 } + 1$ B) $\sqrt { 3 } + 2$ C) $\sqrt { 5 } + 1$ D) $3 - \sqrt { 2 }$