Eccentricity of Hyperbola

Q.

Latus rectum of a hyperbola is 8 and its conjugate axis is equal to half the distance between the foci. What is the eccentricity of the hyperbola.

1. $\frac{4}{3}$

2. $\frac{4}{\sqrt{3}}$

3. None of these

4. $\frac{2}{\sqrt{3}}$

Q.

$e_1$ and $e_2$ are the eccentricities of two conics S and $S^1$. If ${e_1}^2+{e_2}^2$ = 3 then both S and $S^1$ can be

1. ellipse

2. parabolas

3. hyperbolas

4. circles

Q.

If the eccentricities of the hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1and\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$ be e and $e_1$, then $\frac{1}{e^2}+\frac{1}{e_1^2}=$

1. 1

2. 2

3. 3

4. None of these

Q.

e and $e^1$ are the eccentricities of the hyperbolas $16x^2-9y^2=144$ and $9x^2-16y^2$= - 144 then e - $e^1$ =

1. 3/2

2. 2

3. 1

4. 0

Q. 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. $\sqrt{3}$

Q. Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be reciprocal to that of the ellipse $x^2+4y^2=4$. If the hyperbola passes through a focus of the ellipse then

1. the equation of the hyperbola is $\frac{x^4}{5}=\frac{y^1}{2}=1$
2. a focus of the hyperbola is (2, 0)
3. the eccentricity of the hyperbola is $\frac{\sqrt{5}}{3}$
4. the equation of the hyperbola is $x^2-3y^2=3$

Q.

If $e_1$ is the eccentricity of the ellipse $\frac{x^2}{16}+\frac{y^2}{25}=1$ and $e_2$ is the eccentricity of the hyperbola passing through the foci of the ellipse and $e_1{e}_2=1$, then equation of the hyperbola is

1. None of these

2. $\frac{x^2}{9}-\frac{y^2}{16}=1$

3. $\frac{x^2}{16}-\frac{y^2}{9}=-1$

4. $\frac{x^2}{9}-\frac{y^2}{25}=1$

Q. The equations of the directrices of the hyperbola $16x^2-9y^2=-144$ are:

1. $y=\pm \frac{8}{25}$
2. $y=\pm \frac{16}{25}$
3. $y=\pm \frac{8}{5}$
4. $y=\pm \frac{16}{5}$

Q. Find the eccentricity of a hyperbola whose latus rectum is half of its transverse axis.

1. $\frac{\sqrt{5}}{2}$
2. $\sqrt{\frac{3}{2}}$
3. $\sqrt{3}$
4. $\sqrt{2}$

Q. If $e_1$ is the eccentricity of the ellipse $\frac{x^2}{16}+\frac{y^2}{25}=1$ and $e_2$ is the eccentricity of the hyperbola passing through the foci of the ellipse and $e_1.e_2=1$, then the equation of the hyperbola is

1. $\frac{16x^2}{81}-\frac{y^2}{9}=-1$
2. $\frac{16x^2}{81}-\frac{y^2}{9}=1$
3. $\frac{x^2}{9}-\frac{16y^2}{81}=-1$
4. $\frac{x^2}{9}-\frac{16y^2}{81}=1$