Latus Rectum of Ellipse

Q.

The length of the latus rectum of the ellipse $5x^2+9y^2=45$ is

1. 

2. 

3. 

4. 

Q.

Area of the rectangle formed by the ends of latusrecta of the Ellipse $4x^2+9y^2$ = 144 is

1. 

2. 

3. 

4. 

Q. The latusrectum LL’ subtends a right angle at the centre of the ellipse, then its eccentricity is

1. 
2. 
3. 
4. 

Q. If the distance between the foci of an ellipse is $6$ and the distance between its directrices is $12$, then the length of its latus rectum is

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

Q. Let $0<\theta <\frac{\pi }{2}$. If the eccentricity of the hyperbola $\frac{x^2}{{\cos }^2\theta }-\frac{y^2}{{\sin }^2\theta }=1$ is greater than $2$, then the length of its latus rectum lies in the interval:

1. $(1,\frac{3}{2}]$
2. $(\frac{3}{2},2]$
3. $(2,3]$
4. $(3,\infty )$

Q. Let $S$ and $S^{\prime }$ be foci of an ellipse and $B$ be any one of the extremities of its minor axis. If $\Delta S^{\prime }BS$ is a right angled triangle with right angle at $B$ and area of $△S^{\prime }BS=8\text{sq. units}$, then the length of a latus rectum of the ellipse (in units) is :

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

Q. Which of the following is INCORRECT about the ellipse $x^2+4y^2-2x-16y+13=0?$

1. The latus rectum of the ellipse is $1$
2. The distance between foci of the ellipse is $2\sqrt{3}$
3. Eccentricity of the ellipse is $\frac{\sqrt{3}}{2}$
4. Sum of the focal distances of a point $P(x,y)$ on the ellipse is $4\sqrt{3}$

Q. If a variable tangent of the circle $x^2+y^2=1$ intersects the ellipse $x^2+2y^2=4$ at points $P$ and $Q,$ then the locus of the point of intersection of tangent at $P$ and $Q$ is

1. a circle of radius $2$ units
2. a parabola with focus at $(2,3)$
3. an ellipse with latus rectum $2$ units
4. a hyperbola with eccentricity $\frac{3}{2}$

Q. For the horizontal ellipse, the difference between the length of the major axis and the latus rectum of an ellipse is

1. $ae$
2. $2ae$
3. $a{e}^2$
4. $2a{e}^2$

Q. The latus rectum subtends a right angle at the centre of the ellipse, then its eccentricity is

1. $2\sin {18}^{\circ }$
2. $2\cos {18}^{\circ }$
3. $2\sin {54}^{\circ }$
4. $2\cos {54}^{\circ }$