Latus Rectum of Ellipse

Q.

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

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

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

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

4. $\frac{10}{3}$

Q.

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

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

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

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

4. $\frac{10}{3}$

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

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

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

Q. If the length of the latus rectum of an ellipse is $4$ units and the distance between a focus and its nearest vertex on the major axis is $\frac{3}{2}$ units, then its eccentricity is

1. $\frac{2}{3}$
2. $\frac{1}{2}$
3. $\frac{1}{9}$
4. $\frac{1}{3}$

Q. A point moves so that its distance from the point $(2,0)$ is always $\frac{1}{3}$ of its distance from the line $x-18=0.$ If the locus of the point is a conic, then

1. Locus of the point will be $\frac{x^2}{36}+\frac{y^2}{32}=1.$
2. Locus of the point will be $\frac{x^2}{18}+\frac{y^2}{16}=1.$
3. Length of the latus rectum of the conic $=\frac{32}{3}$ units
4. Length of the latus rectum of the conic $=\frac{16\sqrt{2}}{3}$ units

Q.

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

1. $\frac{32\sqrt{5}}{3}$

2. $\frac{64\sqrt{5}}{3}$

3. $\frac{16\sqrt{5}}{3}$

4. $\frac{32\sqrt{3}}{5}$

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