Directrix of Hyperbola

Q. The ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and hyperbola $\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$ are having a same foci and length of minor axis of ellipse is same as the conjugate axis of the hyperbola. If $e_1$ & $e_2$ are the eccentricities of ellipse and hyperbola respectively, then the value of $\frac{1}{e_1^2}+\frac{1}{e_2^2}$ is

1. $1$
2. $2$
3. $3$
4. $4$

Q. The distance between the foci of a hyperbola is double the distance between its vertices and the length of its conjugate axis is $6$. Then which of the following is/are correct?

1. length of transverse axis $=\sqrt{3}$ unit
2. length of latus rectum $=3\sqrt{3}$ unit
3. length of transverse axis $=2\sqrt{3}$ unit
4. length of latus rectum $=6\sqrt{3}$ unit

Q. The foci of the hyperbola are $S(5,6),S^{\prime }(-3,-2)$. If its eccentricity is $2$, then the equation of its directrix corresponding to focus $S$ is

1. $x+y-3=0$
2. $x+y-5=0$
3. $x+y-7=0$
4. $x+y-1=0$

Q. The foci of the hyperbola are $S(5,6),S^{\prime }(-3,-2)$. If its eccentricity is $2$, then the equation of its directrix corresponding to focus $S$ is

1. $x+y-3=0$
2. $x+y-5=0$
3. $x+y-7=0$
4. $x+y-1=0$

Q. Find the coordinates of the foci, the vertices, the lenght of major axis, the minor axis, the eccentricity, and the latus rectum of the ellipse.
$\frac{x^2}{25}+\frac{y^2}{9}=1$

Q. If a directrix of a hyperbola centred at the origin and passing through the point $(4,-2\sqrt{3})$ is $5x=4\sqrt{5}$ and its eccentricity is $e$, then :

1. $4e^4-24e^2+27=0$
2. $4e^4-12e^2-27=0$
3. $4e^4+8e^2-35=0$
4. $4e^4-24e^2+35=0$

Q. The ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and hyperbola $\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$ are having a same foci and length of minor axis of ellipse is same as the conjugate axis of the hyperbola. If $e_1$ & $e_2$ are the eccentricities of ellipse and hyperbola respectively, then the value of $\frac{1}{e_1^2}+\frac{1}{e_2^2}$ is

1. $2$
2. $3$
3. $1$
4. $4$

Q. If $a,b,c$ are in $A.P$, the straight line $ax+by+c=0$, passes through a fixed point which lie on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{4}=1,$ then eccentricity of hyperbola is

Q. The locus of the point of intersection of the lines $bxt-ayt=ab$ and $bx+ay=abt$, $t$ being parameter is

1. an ellipse with equation $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$
2. an ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
3. a hyperbola with equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
4. a hyperbola with equation $\frac{x^2}{b^2}-\frac{y^2}{a^2}=1$

Q.

In a hyperbola e = $\frac{9}{4}$ and the distance between the directrices is 3. Then the length of Transverse axis is

1. $\frac{27}{2}$

2. $\frac{27}{8}$

3. $\frac{27}{4}$

4. $\frac{17}{4}$

Q.

From a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ lines are drawn to focus S

and directrix perpendicular to it as shown.Then,

1. $PS>PD$

2. $PS=PD$

3. $PS<PD$

4. \(No~such~strict~relation\)

Q.

In a hyperbola the distance between the foci is 2 and the distance between directrices is 1 Its eccentricity is

1. $\sqrt{3}$

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

3. $\sqrt{2}$

4. $\frac{5}{2}$

Q. The ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and hyperbola $\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$ are having a same foci and length of minor axis of ellipse is same as the conjugate axis of the hyperbola. If $e_1$ & $e_2$ are the eccentricities of ellipse and hyperbola respectively, then the value of $\frac{1}{e_1^2}+\frac{1}{e_2^2}$ is

1. $2$
2. $4$
3. $3$
4. $1$