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) $x^{2}+4y^{2}=16$

If the eccentricity of a hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, which passes through $( K , 2 )$, is $\frac { \sqrt { 13 } } { 3 }$, then the value of $K ^ { 2 }$ is
(1) 18
(2) 8
(3) 1
(4) 2

A chord is drawn through the focus of the parabola $y ^ { 2 } = 6 x$ such that its distance from the vertex of this parabola is $\frac { \sqrt { 5 } } { 2 }$, then its slope can be
(1) $\frac { \sqrt { 5 } } { 2 }$
(2) $\frac { 2 } { \sqrt { 3 } }$
(3) $\frac { \sqrt { 3 } } { 2 }$
(4) $\frac { 2 } { \sqrt { 5 } }$

The tangent at an extremity (in the first quadrant) of the latus rectum of the hyperbola $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 5 } = 1$, meets the $x$-axis and $y$-axis at $A$ and $B$, respectively. Then $OA ^ { 2 } - OB ^ { 2 }$, where $O$ is the origin, equals
(1) $- \frac { 20 } { 9 }$
(2) $\frac { 16 } { 9 }$
(3) 4
(4) $- \frac { 4 } { 3 }$

If $OB$ is the semi-minor axis of an ellipse, $F _ { 1 }$ and $F _ { 2 }$ are its focii and the angle between $F _ { 1 } B$ and $F _ { 2 } B$ is a right angle, then the square of the eccentricity of the ellipse is
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { \sqrt { 2 } }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 2 \sqrt { 2 } }$

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is: (1) $\frac{4}{3}$ (2) $\frac{4}{\sqrt{3}}$ (3) $\frac{2}{\sqrt{3}}$ (4) $\sqrt{3}$

$P$ and $Q$ are two distinct points on the parabola, $y ^ { 2 } = 4 x$, with parameters $t$ and $t _ { 1 }$, respectively. If the normal at $P$ passes through $Q$, then the minimum value of $t _ { 1 } ^ { 2 }$, is
(1) 8
(2) 4
(3) 6
(4) 2

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) $( \sqrt { 10 } , 2 \sqrt { 3 } )$

The eccentricity of an ellipse whose centre is at the origin is $\dfrac{1}{2}$. If one of its directrices is $x = -4$, then the equation of the normal to it at $\left(1, \dfrac{3}{2}\right)$ is:
(1) $2y - x = 2$
(2) $4x - 2y = 1$
(3) $4x + 2y = 7$
(4) $x + 2y = 4$

A hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $(\pm 2, 0)$. Then the tangent to this hyperbola at $P$ also passes through the point
(1) $(3\sqrt{2}, 2\sqrt{3})$
(2) $(2\sqrt{2}, 3\sqrt{3})$
(3) $(\sqrt{3}, \sqrt{2})$
(4) $(-\sqrt{2}, -\sqrt{3})$

A hyperbola passes through the point $P ( \sqrt { 2 } , \sqrt { 3 } )$ and has foci at $( \pm 2 , 0 )$. Then the tangent to this hyperbola at $P$ also passes through the point:
(1) $( 3 \sqrt { 2 } , 2 \sqrt { 3 } )$
(2) $( 2 \sqrt { 2 } , 3 \sqrt { 3 } )$
(3) $( \sqrt { 3 } , \sqrt { 2 } )$
(4) $( - \sqrt { 2 } , - \sqrt { 3 } )$

The locus of the point of intersection of the lines $\sqrt { 2 } x - y + 4 \sqrt { 2 } k = 0$ and $\sqrt { 2 } k x + k y - 4 \sqrt { 2 } = 0$ ( $k$ is any non-zero real parameter) is
(1) an ellipse whose eccentricity is $\frac { 1 } { \sqrt { 3 } }$
(2) a hyperbola whose eccentricity is $\sqrt { 3 }$
(3) a hyperbola with length of its transverse axis $8 \sqrt { 2 }$
(4) an ellipse with length of its major axis $8 \sqrt { 2 }$

If $\beta$ is one of the angles between the normals to the ellipse $x ^ { 2 } + 3 y ^ { 2 } = 9$ at the points $( 3 \cos \theta , \sqrt { 3 } \sin \theta )$ and $( - 3 \sin \theta , \sqrt { 3 } \cos \theta ) ; \theta \in \left( 0 , \frac { \pi } { 2 } \right) ;$ then $\frac { 2 \cot \beta } { \sin 2 \theta }$ is equal to :
(1) $\frac { 1 } { \sqrt { 3 } }$
(2) $\frac { \sqrt { 3 } } { 4 }$
(3) $\frac { 2 } { \sqrt { 3 } }$
(4) $\sqrt { 2 }$

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is $\frac { 3 } { 2 }$ units, then its eccentricity is
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { 9 }$
(4) $\frac { 1 } { 3 }$

If the tangent drawn to the hyperbola $4 y ^ { 2 } = x ^ { 2 } + 1$ intersect the co-ordinates axes at the distinct points $A$ and $B$, then the locus of the midpoint of $AB$ is :
(1) $x ^ { 2 } - 4 y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(2) $4 x ^ { 2 } - y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(3) $x ^ { 2 } - 4 y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
(4) $4 x ^ { 2 } - y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$

If $\beta$ is one of the angles between the normals to the ellipse, $x ^ { 2 } + 3 y ^ { 2 } = 9$ at the points ( $3 \cos \theta , \sqrt { 3 } \sin \theta$ ) and $( - 3 \sin \theta , \sqrt { 3 } \cos \theta ) ; \in \left( 0 , \frac { \pi } { 2 } \right) ;$ then $\frac { 2 \cot \beta } { \sin 2 \theta }$ is equal to
(1) $\sqrt { 2 }$
(2) $\frac { 2 } { \sqrt { 3 } }$
(3) $\frac { 1 } { \sqrt { 3 } }$
(4) $\frac { \sqrt { 3 } } { 4 }$

Tangents are drawn to the hyperbola $4 x ^ { 2 } - y ^ { 2 } = 36$ at the points $P$ and $Q$. If these tangents intersect at the point $T ( 0,3 )$ then the area (in sq. units) of $\triangle P T Q$ is:
(1) $36 \sqrt { 5 }$
(2) $45 \sqrt { 5 }$
(3) $54 \sqrt { 3 }$
(4) $60 \sqrt { 3 }$

If the tangents drawn to the hyperbola $4 y ^ { 2 } = x ^ { 2 } + 1$ intersect the co-ordinate axes at the distinct points $A$ and $B$, then the locus of the mid point of $A B$ is
(1) $x ^ { 2 } - 4 y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(2) $4 x ^ { 2 } - y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(3) $4 x ^ { 2 } - y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
(4) $x ^ { 2 } - 4 y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$

The tangent to the parabola $y ^ { 2 } = 4 x$ at the point where it intersects the circle $x ^ { 2 } + y ^ { 2 } = 5$ in the first quadrant, passes through the point:
(1) $\left( \frac { 1 } { 4 } , \frac { 3 } { 4 } \right)$
(2) $\left( - \frac { 1 } { 3 } , \frac { 4 } { 3 } \right)$
(3) $\left( - \frac { 1 } { 4 } , \frac { 1 } { 2 } \right)$
(4) $\left( \frac { 3 } { 4 } , \frac { 7 } { 4 } \right)$

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $( 0 , 5 \sqrt { 3 } )$, then the length of its latus rectum is:
(1) 6
(2) 10
(3) 8
(4) 5

If the parabolas $y ^ { 2 } = 4 b ( x - c )$ and $y ^ { 2 } = 8 a x$ have a common normal, then which one of the following is a valid choice for the ordered triad $( a , b , c )$
(1) $( 1,1,3 )$
(2) $\left( \frac { 1 } { 2 } , 2,0 \right)$
(3) $\left( \frac { 1 } { 2 } , 2,3 \right)$
(4) All of above

If the eccentricity of the standard hyperbola passing through the point $( 4 , 6 )$ is 2 , then the equation of the tangent to the hyperbola at $( 4 , 6 )$ is:
(1) $2 x - 3 y + 10 = 0$
(2) $x - 2 y + 8 = 0$
(3) $3 x - 2 y = 0$
(4) $2 x - y - 2 = 0$

If the line $y = m x + 7 \sqrt { 3 }$ is normal to the hyperbola $\frac { x ^ { 2 } } { 24 } - \frac { y ^ { 2 } } { 18 } = 1$, then a value of $m$ is:
(1) $\frac { \sqrt { 5 } } { 2 }$
(2) $\frac { 3 } { \sqrt { 5 } }$
(3) $\frac { \sqrt { 15 } } { 2 }$
(4) $\frac { 2 } { \sqrt { 5 } }$

The equation of a tangent to the hyperbola, $4 x ^ { 2 } - 5 y ^ { 2 } = 20$, parallel to the line $x - y = 2$, is
(1) $x - y + 7 = 0$
(2) $x - y - 3 = 0$
(3) $x - y + 1 = 0$
(4) $x - y + 9 = 0$

If the tangent to the parabola $y ^ { 2 } = x$ at a point $( \alpha , \beta ) , ( \beta > 0 )$ is also a tangent to the ellipse, $x ^ { 2 } + 2 y ^ { 2 } = 1$ then $\alpha$ is equal to:
(1) $\sqrt { 2 } - 1$
(2) $2 \sqrt { 2 } + 1$
(3) $\sqrt { 2 } + 1$
(4) $2 \sqrt { 2 } - 1$