The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is

(a)

\(\frac { 4 }{ 3 } \)

(b)

\(\frac { 4 }{ \sqrt { 3 } } \)

(c)

\(\frac { 2 }{ \sqrt { 3 } } \)

(d)

\(\frac { 3 }{ 2 } \)

The area of quadrilateral formed with foci of the hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text { and } \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\)

(a)

4(a²+b²)

(b)

2(a²+b²)

(c)

a² +b²

(d)

\(\frac { 1 }{ 2 } \)(a²+b²)

Tangents are drawn to the hyperbola  \(\frac { { x }^{ 2 } }{ 9 } -\frac { { y }^{ 2 } }{ 4 } =1\) parallel to the straight line 2x − y = 1. One of the points of contact of tangents on the hyperbola is

(a)

\(\left(\frac{9}{2 \sqrt{2}}, \frac{-1}{\sqrt{2}}\right)\)

(b)

\(\left(\frac{-9}{2 \sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)

(c)

\(\left(\frac{9}{2 \sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)

(d)

\((3 \sqrt{3},-2 \sqrt{2})\)

Area of the greatest rectangle inscribed in the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is

(a)

2ab

(b)

ab

(c)

\( \sqrt{ ab}\)

(d)

\(\frac { a }{ b } \)

The values of m for which the line y = mx + \(2\sqrt { 5 } \) touches the hyperbola 16x² − 9y² = 144 are the roots of x² − (a + b)x − 4 = 0, then the value of (a+b) is

(a)

2

(b)

4

(c)

0

(d)

-2

The equation of the directrix of the parabola y²+ 4y + 4x + 2 = 0 is ___________

(a)

x = -1

(b)

x = 1

(c)

x = \(\frac{-3}{2}\)

(d)

x = \(\frac{3}{2}\)

If the distance between the foci is 2 and the distance between the direction is 5, then the equation of the ellipse is __________

(a)

6x² + 10y² = 5

(b)

6x² + 10y² = 15

(c)

x² + 3y² = 10

(d)

none

The length of major and minor axes of 4x² + 3y² = 12 are ____________

(a)

4, 2\(\sqrt3\)

(b)

2, \(\sqrt3\)

(c)

2\(\sqrt3\), 4

(d)

\(\sqrt3\), 2

The number of normals that can be drawn from a point to the parabola y² = 4ax is __________

(a)

3

(b)

2

(c)

0

(d)

1

The locus of the point of intersection of perpendicular tangents of the parabola y² = 4ax is

(a)

latus rectum

(b)

directrix

(c)

tangent at the vertex

(d)

axis of the parabola