An ellipse is drawn by taking a diameter of the circle $(x-1)^{2}+y^{2}=1$ as its semi-minor axis and a diameter of the circle $x^{2}+(y-2)^{2}=4$ as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the ellipse is
(1) $4x^{2}+y^{2}=4$
(2) $x^{2}+4y^{2}=8$
(3) $4x^{2}+y^{2}=8$
(4) $x^{2}+4y^{2}=16$

A hyperbola whose transverse axis is along the major axis of the conic $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 4 } = 4$ and has vertices at the foci of the conic. If the eccentricity of the hyperbola is $\frac { 3 } { 2 }$, then which of the following points does not lie on the hyperbola?
(1) $( \sqrt { 5 } , 2 \sqrt { 2 } )$
(2) $( 0,2 )$
(3) $( 5,2 \sqrt { 3 } )$
(4) $( \sqrt { 10 } , 2 \sqrt { 3 } )$

If $3 x + 4 y = 12 \sqrt { 2 }$ is a tangent to the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 9 } = 1$ for some $a \in R$, then the distance between the foci of the ellipse is
(1) $2 \sqrt { 7 }$
(2) 4
(3) $2 \sqrt { 5 }$
(4) $2 \sqrt { 2 }$

A hyperbola having the transverse axis of length, $\sqrt { 2 }$ has the same foci as that of the ellipse, $3 x ^ { 2 } + 4 y ^ { 2 } = 12$ then this hyperbola does not pass through which of the following points?
(1) $\left( \frac { 1 } { \sqrt { 2 } } , 0 \right)$
(2) $\left( - \sqrt { \frac { 3 } { 2 } } , 1 \right)$
(3) $\left( 1 , - \frac { 1 } { \sqrt { 2 } } \right)$
(4) $\left( \sqrt { \frac { 3 } { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$

Let $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b )$ be a given ellipse, length of whose latus rectum is 10 . If its eccentricity is the maximum value of the function, $\phi ( t ) = \frac { 5 } { 12 } + t - t ^ { 2 }$, then $a ^ { 2 } + b ^ { 2 }$ is equal to:
(1) 145
(2) 116
(3) 126
(4) 135

A hyperbola passes through the foci of the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:
(1) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$
(2) $x ^ { 2 } - y ^ { 2 } = 9$
(3) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$
(4) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$

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

Q66. 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

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) $3 - \sqrt { 2 }$