Latus Rectum of Hyperbola

Q. The point of intersection of two tangents to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ the product of whose slopes is $c^2$, lies on the curve.

1. 
   
2. 
   
3. 
   
4. 
   

Q.

For some$\theta \in (0,\frac{\mathrm{\pi }}{2})$, if the eccentricity of the hyperbola, $x^2-y^2{\sec }^2\left(\theta \right)=10$ is $\sqrt{5}$ times the eccentricity of the ellipse, $x^2{\sec }^2\left(\theta \right)+y^2=5$, then the length of the latus rectum of the ellipse, is:

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

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

3. $2\sqrt{6}$

4. $\sqrt{30}$

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

1. 
2. 
3. 
4. 

Q. Let $P$ be any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that the absolute difference of the distances of $P$ from the two foci is $12.$ If the eccentricity of the hyperbola is $2,$ then the length of the latus rectum is

1. $4\sqrt{3}$ unit
2. $18$ unit
3. $2\sqrt{3}$ unit
4. $36$ unit

Q.

The equation of the hyperbola having its eccentricity $2$ and the distance between its foci is $8$, is:

1. $\frac{x^2}{12}-\frac{y^2}{4}=1$

2. $\frac{x^2}{4}-\frac{y^2}{12}=1$

3. $\frac{x^2}{8}-\frac{y^2}{2}=1$

4. $\frac{x^2}{16}-\frac{y^2}{9}=1$

Q. Equation of one of the latus rectum of the hyperbola $(10x-5{)}^2+(10y-2{)}^2=9(3x+4y-7{)}^2$ is

1. $30x+40y-23=0$
2. $30x+40y+23=0$
3. $40x+30y-23=0$
4. $40x+30y+23=0$

Q. The area of the figure bounded by the parabolas $x=-2y^2$ and $x=1-3y^2$ is

1. $\frac{4}{3}\text{square units}$
2. $\frac{2}{3}\text{square units}$
3. $\frac{3}{7}\text{square units}$
4. $\frac{6}{7}\text{square units}$

Q. The equation of the latus rectum of the parabola represented by equation $y^2+2Ax+2By+C=0$ is

1. 
   
2. 
3. 
4. 

Q.

$\text{The latus-rectum of the hyperbola}$

$16x^2-9y^2=144is$

1. 16/3

2. 32/3

3. 8/3

4. 4/3

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. If the coordinates of the vertices of a hyperbola are $(9,2)$ and $(1,2)$ and the distance between its foci is $10$ units. Then

1. length of its latus rectum is $9$ units
2. length of its latus rectum is $\frac{9}{2}$ units
3. equation of the hyperbola is $\frac{(x-5{)}^2}{9}$- \dfrac{(y-2)^2}{16}=1$
4. equation of the hyperbola is $\frac{(x-5{)}^2}{16}$- \dfrac{(y-2)^2}{9}=1$

Q. Find the equation of ellipse whose eccentricity is $\frac{2}{3}$, latus rectum is $5$ and the centre is $(0,0)$.

Q. An ellipse intersects the hyperbola $2x^2-2y^2=1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinates axes, then

1. the foci of ellipse are $(\pm \sqrt{2},0)$
2. Equation of ellipse is $x^2+2y^2=1$
3. the foci of ellipse are $(\pm 1,0)$
4. equation of ellipse are $x^2+2y^2=4$

Q. Let $P$ be any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that the absolute difference of the distances of $P$ from the two foci is $12.$ If the eccentricity of the hyperbola is $2,$ then the length of the latus rectum is

1. $4\sqrt{3}$ unit
2. $18$ unit
3. $2\sqrt{3}$ unit
4. $36$ unit

Q. The locus of the point of interaction of two tangents of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ which make an angle α with one another is

1. 
2. 
3. 
4. 

Q.

If a hyperbola has a length of its conjugate axis equal to $5$ and the distance between its foci is $13$, then the eccentricity of the hyperbola is:

1. $9$

2. $\frac{13}{12}$

3. $\frac{32}{3}$

4. $\frac{64}{3}$

Q. If focii of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ coincide with the focii of $\frac{x^2}{25}+\frac{y^2}{9}=1$ and eccentricity of the hyperbola is $2$, then

1. $a^2+b^2=14$
2. There is a director circle of the hyperbola
3. Length of latus rectum of the hyperbola is $12$
4. Centre of the director circle is $(0,0)$

Q.

$\text{Write the length of the latus-rectum of the hyperbola}16x^2-9y^2=144$

Q. Let $a+b+c=10$, $a,b,c>0$. Then the maximum value of $a^2{b}^3{c}^5$ is

1. ${\left(15\right)}^3\times 100$
2. ${\left(15\right)}^3\times 10$
3. ${\left(15\right)}^3\times 1000$
4. ${\left(60\right)}^3$

Q. The solution of $\frac{dy}{dx}=\frac{x^2+y^2+1}{2xy}$ satisfying y(1) = 0 is

1. an ellipse with e $=\frac{1}{\sqrt{2}}$
2. a hyperbola with e = $\sqrt{2}$
3. a circle whose radius is 1
4. a curve symmetric about origin

Q. A variable plane forms a tetrahedron of constant volume $64K^3$ with the coordinate planes and the origin. The locus of the centroid of the tetrahedron is

1. $xyz=6k^3$
2. $x^3+y^3+z^3=6K^2$
3. $x^2+y^2+z^2=4K^2$
4. $x^{-2}+y^{-2}+z^{-2}=4k^{-2}$

Q. A plane intersects the co ordinate axes at $A,B,C$. If $O=(0,0,0)$ and $(1,1,1)$ is the centroid of the tetrahedron $OABC$, then the sum of the reciprocals of the intercepts of the plane

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

Q. The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ has its conjugate axis of length $5$ and passes through the point $(2,1)$. The length of latus rectum is :

1. $\frac{5}{4}\sqrt{29}$
2. $\frac{5}{8}\sqrt{29}$
3. $\frac{\sqrt{29}}{4}$
4. $\sqrt{29}$

Q. Find the vertex, axis, focus, directrix and latus rectum of the parabola, also draw their rough sketches
$4y^2+12x-20y+67=0$

Q.

The area (in square units) of the quadrilateral formed by the two pairs of lines

$l^2{x}^2-m^2{y}^2-n(lx+my)=0$

and $l^2{x}^2-m^2{y}^2+n(lx-my)=0$ is

1. 

2. 

3. 

4. 

Q. Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be reciprocal to that of the ellipse $x^2+4y^2=4$. lf the hyperbola passes through a focus of the ellipse, then

1. the equation of the hyperbola is $\frac{x^2}{3}-\frac{y^2}{2}=1$
2. a focus of the hyperbola is $(2,0)$
3. the equation of the hyperbola is $x^2-3y^2=3$
4. the eccentricity of the hyperbola is $\sqrt{\frac{5}{3}}$

Q. Equation of one of the latus rectum of the hyperbola $(10x-5{)}^2+(10y-2{)}^2=9(3x+4y-7{)}^2$ is

1. $30x+40y-23=0$
2. $40x+30y-23=0$
3. $30x+40y+23=0$
4. $40x+30y+23=0$

Q. If the coordinates of the vertices of a hyperbola are $(9,2)$ and $(1,2)$ and the distance between its foci is $10$ units. Then

Q.

If the latus rectum of an hyperbola be 8 and eccentricity be $\frac{3}{\sqrt{5}}$ , then the equation of the hyperbola is

1. $4x^2-5y^2=100$

2. $5x^2-4y^2=100$

3. $4x^2+5y^2=100$

4. $5x^2+4y^2=100$

Q. The eccentricity of the hyperbola whose latus-return is $8$ and length of the conjugate axis is equal to half the distance between the foci, is

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