Latus Rectum of Ellipse

Q.

All possible values of $\theta \in \left[0,2\mathrm{\pi }\right]$ for which $\sin 2\theta +\tan 2\theta >0$ lie in

1. $\left(0,\left(\frac{\pi }{2}\right)\right)\cup \left(\pi ,\frac{3\pi }{2}\right)$

2. $\left(0,\frac{\pi }{4}\right)\cup \left(\frac{\pi }{2},\frac{3\pi }{4}\right)\cup \left(\pi ,\frac{5\pi }{4}\right)\cup \left(\frac{3\pi }{2},\frac{7\pi }{4}\right)$

3. $\left(0,\frac{\mathrm{\pi }}{2}\right)\cup \left(\frac{\mathrm{\pi }}{2}\frac{3\mathrm{\pi }}{4}\right)\cup \left(\mathrm{\pi },\frac{7\mathrm{\pi }}{6}\right)$

4. $\left(0,\frac{\mathrm{\pi }}{4}\right)\cup \left(\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{4}\right)\cup \left(\frac{3\mathrm{\pi }}{2},\frac{11\mathrm{\pi }}{6}\right)$

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. Statement-$1$: An equation of a common tangent to the parabola
$y^2=16\sqrt{3}x$ and the ellipse $2x^2+y^2=4$ is $y=2x+2\sqrt{3}$

Statement-$2$: If the line $y=mx+\frac{4\sqrt{3}}{m},(m\ne 0)$ is a common tangent to the parabola $y²=16\sqrt{3}x$ and the ellipse $2x^2+y^2=4$ then $m$ satisfies:
$m^4+2m^2=24.$

1. Statement-$1$ is false, Statement-$2$ is true.
2. Statement-$1$ is true, statement-$2$ is true; statement-$2$ is a correct explanation for Statement-$1$.
3. Statement-$1$ is true, statement-$2$ is true; statement-$2$ is not a correct explanation for Statement-$1$.
4. Statement-$1$ is true, statement-$2$ is false.

Q. $S$ and $T$ are the foci of an ellipse and $B$ is the end point of the minor axis. If $STB$ is equilateral triangle, the eccentricity of the ellipse is

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

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. The latus-rectum of the conic $3x^2+4y^2-6x+8y-5=0$ is

1. 3
2. $\frac{\sqrt{3}}{2}$
3. $\frac{2}{\sqrt{3}}$
4. none of these

Q.

If in a $△ABC,\sin A,\sin B,$ and $\sin C$ are in AP, then

1. the radius are in AP

2. the altitudes are in HP

3. the angles are in AP

4. the angles are in HP

Q. The difference between the lengths of the major axis and the latus-rectum of an ellipse is

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

Q.

If the centre, one of the foci and semi-major axis of an ellipse be $(0,0),(0,3)$and $5$ , then its equation is:

1. $\frac{x^2}{16}+\frac{y^2}{25}=1$

2. $\frac{x^2}{25}+\frac{y^2}{16}=1$

3. $\frac{x^2}{9}+\frac{y^2}{25}=1$

4. none of these

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. If $m_1$ and $m_2$ are the slopes of the tangents drawn from the point $P(6,-2)$ to the ellipse $4x^2+9y^2=36,$ then $m_1^2+m_2^2$ is equal to

1. $\frac{16}{9}$
2. $\frac{32}{27}$
3. $\frac{64}{81}$
4. $\frac{32}{9}$

Q.

If eccentricity of the hyperbola

$\frac{x^2}{{\cos }^2\theta }-\frac{y^2}{{\sin }^2\theta }=1$

is more than $2$ when $\in (0,\pi /2)$ then values of length of latus rectum lies in the interval

1. $(3,\infty )$

2. $\left(1,\frac{3}{2}\right)$

3. $\left(2,3\right)$

4. $(-3,-2)$

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. 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. 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. Find the latus rectum of the ellipse: $5x^2+4y^2=1$

1. $\frac{2}{5}$
2. $\frac{8}{5}$
3. $\frac{4}{5}$
4. $\frac{12}{5}$

Q.

A circle is centered at origin. 2 points P and Q lies on the positive x axis outside the circle such that OQ = 2OP. The length of tangent drawn from Q to the circle is thrice the length of tangent from P to the circle. What the radius of circle if OP = a.

1. 

2. 

3. 

4. 

Q.

Given standard equation of ellipse,
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$,
with eccentricity $e$.
Match the following
$\begin{array}{cc}a)\text{Major axis}& i\left)2a\right(1-e^2)\\ b)\text{Minor axis}& ii)y=0\\ c)\text{Double ordinate}& iii)x=0\\ d)\text{Latus Rectum length}& iv)x=\frac{-a}{e}\\ & v)\sqrt{1-\frac{b^2}{a^2}}\end{array}$

1. $a=ii,b=iii,c=iii,d=i$

2. $a=v,b=ii,c=vi,d=i$

3. $a=ii,b=iv,c=iii,d=v$

4. $a=iii,b=ii,c=vi,d=i$

Q.

If $\theta ={\sin }^{-1}x+{\cos }^{-1}x-{\tan }^{-1}x,x\ge 0$, then the smallest interval in which $\theta$ lies is given by:

1. $\frac{\mathrm{\pi }}{2}\le \theta \le \frac{3\mathrm{\pi }}{4}$

2. $0<\theta <\mathrm{\pi }$

3. $-\frac{\mathrm{\pi }}{4}\le \theta \le 0$

4. $\frac{\mathrm{\pi }}{4}\le \theta \le \frac{\mathrm{\pi }}{2}$

Q. If a variable tangent to the circles $x^2+y^2=1$ intersects the ellipse $x^2+{2y}^2=4$ at points P and Q then the locus of the point of intersection of tangents to the ellipse at P and Q is a conic whose

1. eccentricity $\frac{\sqrt{3}}{4}$
2. eccentricity is $\frac{\sqrt{5}}{2}$
3. latus-rectum is of length 2 units
4. foci are $(\pm 2\sqrt{5},0)$

Q. The eccentricity of the ellipse whose latus rectum is equal to $\frac{3}{2}$ (half of its minor axis) is

1. 
2. 
3. 
4. 

Q. The equation of an ellipse is given in its standard form as $16x^2+y^2=16$, then the length of the latus rectum is,

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

Q.

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

1. 

2. 

3. 

4. 

Q.

The difference between the lengths of the major axis and latus-rectum of an ellipse is

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

4. $ae$

Q. The equation $y^2+4x+4y+k=0$ represents a parabola whose latus rectum is

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

Q. Let $L$ be a tangent line to the parabola $y^2=4x–20$ at $(6,2).$ If $L$ is also a tangent to the ellipse $\frac{x^2}{2}+\frac{y^2}{b}=1$ then the value of $b$ is equal to :

1. $20$
2. $14$
3. $16$
4. $11$

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. The length of the latus rectum of the ellipse $9x^2+4y^2=1$, is

1. $\frac{3}{2}$
2. $\frac{8}{3}$
3. $\frac{4}{9}$
4. $\frac{8}{9}$

Q. If the latus rectum of an ellipse is equal to half of its minor axis, then its eccentricity is

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

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