Latus Rectum of Hyperbola

Q. The length of the Latus rectum of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is.

1. $\frac{-2b^2}{a}$
2. $\frac{b^2}{a}$
3. $\frac{2a^2}{b}$
4. $\frac{2b^2}{a}$

Q. $A$ is the vertex of the hyperbola $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-3=0$, $B$ is one of the end points of latus rectum and $C$ is the focus of the hyperbola. If $A,B$ and $C$ lies on same side of conjugate axis, then the area of the triangle $ABC$ is

1. $2$ sq. units
2. $\sqrt{\frac{3}{2}}-1$ sq. units
3. $1-\sqrt{\frac{2}{3}}$ sq. units
4. $\sqrt{\frac{3}{2}}+1$ sq. units

Q.

In a hyperbola the latusrectum equals to semitransverse axis, then its eccentricity is

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

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

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

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

Q. $A$ is the vertex of the hyperbola $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-3=0$, $B$ is one of the end points of latus rectum and $C$ is the focus of the hyperbola. If $A,B$ and $C$ lies on same side of conjugate axis, then the area of the triangle $ABC$ is

1. $\sqrt{\frac{3}{2}}+1$ sq. units
2. $\sqrt{\frac{3}{2}}-1$ sq. units
3. $1-\sqrt{\frac{2}{3}}$ sq. units
4. $2$ sq. units