Conjugate Hyperbola

Q. The foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the hyperbola $\frac{x^2}{144}+\frac{y^2}{81}=\frac{1}{25}$ coincide. Then the value of $b^2$ is

1. 5
2. 7
3. 9
4. 1

Q.

The eccentricity of the hyperbola $\frac{x^2}{16}-\frac{y^2}{25}=1$ is

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

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

3. $\frac{\sqrt{41}}{4}$

4. $\frac{\sqrt{41}}{5}$

Q. If the angle between the asymptotes of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\frac{\pi }{3}.$ Then the eccentricity of conjugate hyperbola is

Q. If a hyperbola has one focus at the origin and its eccentricity is $\sqrt{2}$ . One of the directrices is $x+y+1=0.$ Then the equations to its asymptotes are

1. 
2. 
3. 
4. 

Q. The eccentricity of the conjugate hyperbola of the hyperbola $x^2-3y^2=1$ is

Q.

If the eccentricity of a hyperbola is $\sqrt{3}$, then the eccentricity of its conjugate hyperbola is

1. $\sqrt{2}$

2. $\sqrt{3}$

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

4. $2\sqrt{3}$

Q.

If $e_1$ is the eccentricity of the ellipse $\frac{x^2}{16}+\frac{y^2}{7}=1$- and $e_2$ is the eccentricity of the hyperbola $\frac{x^2}{9}-\frac{y^2}{7}=1$, then $e_1+e_2=$

1. $\frac{16}{7}$

2. $\frac{25}{4}$

3. $\frac{25}{12}$

4. $\frac{16}{9}$

5. $\frac{23}{16}$

Q. If $S_1$ and $S_2$ are the foci of the hyperbola whose transverse axis length is $4$ units and conjugate axis length is $6$ units, $S_3$ and $S_4$ are the foci of the conjugate hyperbola, then the area of the quadrilateral $S_1{S}_3{S}_2{S}_4$ is sq. units

Q.

The equation of a hyperbola , conjugate to the hyperbola $x^2+3xy+2y^2+2x+3y=0is$

1. 

2. 

3. 

4. 

Q.

The foci of the ellipse $\frac{{\mathrm{x}}^2}{16}+\frac{{\mathrm{y}}^2}{b^2}=1$ and the hyperbola $\frac{{\mathrm{x}}^2}{144}-\frac{{\mathrm{y}}^2}{81}=\frac{1}{25}$ coincide. Then the value of $b^2$ is

1. $1$

2. $5$

3. $7$

4. $9$

Q. The equation of the conjugate hyperbola of the hyperbola $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-3=0$ is

1. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+5=0$
2. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+2=0$
3. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+3=0$
4. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-2=0$

Q. ntFind the scalar and vector components of (3i+j+3k) in the direction of (i-j-k).n

Q. The eccentricity of the conjugate hyperbola of $\frac{x^2}{16}-\frac{y^2}{9}=1$ is .

1. $\frac{5}{4}$
2. $\frac{25}{16}$
3. $\frac{5}{3}$
4. $\frac{25}{9}$

Q. The equation of the hyperbola whose asymptotes are $3x+4y–2=0,2x+y+1=0$ and which passes
through the point (1, 1) 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 $x^2+4y^2=4$. If the hyperbola passes through a focus of the ellipse, then:

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

Q. Find magnitude of p if 2i - j + k, i + 3j - 3k and 3i - pj + 5k are co planar

Q. lf $f\left(x\right)=\frac{\tan x}{\sqrt{1+{\tan }^{2}x}},\underset{x\to (\pi /2{)}^{-}}{Lim}f\left(x\right)=a$ and $\underset{x\to (\pi /2{)}^{+}}{Lim}f\left(x\right)=b$ then

1. $a=b$
2. $a+b=0$
3. $a=1+b$
4. $a+b=2$

Q. The domain of the derivative of the function $f\left(x\right)=\{\begin{array}{cc}{\tan }^{-1}x& for\left|x\right|\le 1\\ \frac{1}{2}(\left|x\right|-1)& for\left|x\right|>1\end{array}$ is

1. $R-\left\{1\right\}$
2. $R-\{-1,1\}$
3. $R-\{-1\}$
4. $R-\left\{0\right\}$

Q.

A hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is drawn along with its conjugate hyperbola. The foci points of both hyperbolas are connected as shown. Then $S_1{S}_3{S}_2{S}_4$ always forms a

1. Rhombus

2. Trapezium

3. Square

4. None of the above

Q. The equation of a hyperbola, conjugate to the hyperbola $x^2+3xy+2y^2+2x+3y+1=0$, is

1. $x^2+3xy+2y^2+2x+3y+1=0$
2. $x^2+3xy+2y^2+2x+3y+2=0$
3. $x^2+3xy+2y^2+2x+3y+3=0$
4. $x^2+3xy+2y^2+2x+3y+4=0$

Q. For hyperbola $\frac{x^2}{16}-\frac{y^2}{12}=0$ and circle $x^2+y^2=25$, find concyclic points.

1. $x=\frac{\pm 10}{\sqrt{7}},y=\frac{\pm 5\sqrt{3}}{\sqrt{7}}$
2. $x=\frac{\pm 7}{\sqrt{1}0},y=\frac{\pm 5\sqrt{3}}{\sqrt{7}}$
3. $x=\frac{\pm 10}{\sqrt{7}},y=\frac{\pm 3\sqrt{5}}{\sqrt{7}}$
4. $x=\frac{\pm 5}{\sqrt{7}},y=\frac{\pm 10\sqrt{3}}{\sqrt{7}}$

Q.

What is the conjugate hyperbola of the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

1. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

2. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

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

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

Q. Consider the parabola $x=ay-b{y}^2$ (where $b\ne 0$) If the exhaustive set of values of a for which there exist $\alpha ,\beta ϵR-\left\{0\right\}$ such that both the point $(\alpha ,\beta )$ and $(\beta ,\alpha )$ lies on the given parabola is $(-\infty ,p)\cup (q,\infty )$ then $\frac{p^2+q^2}{4}$ is

Q. If $a\cdot 3^{tanx}+a\cdot 3^{-tanx}-2=0$ has real solution, $x\ne \frac{\pi }{2},0\le x\le \pi ,$ then the set of possible values of the parameter $a$ are

1. $[-1,1]$
2. $[-1,0)$
3. $(0,1]$
4. $(0,+\infty )$

Q. The equation of a hyperbola, conjugate to the hyperbola $x^2+3xy+2y^2+2x+3y+1=0$, is

1. 
2. 
3. 
4. 

Q. If e and e' be the eccentricities of a hyperbola and its conjugate then
$\frac{1}{e^2}+\frac{1}{(e^{\prime }{)}^2}=1$.

1. True
2. False

Q. The eccentricity of the conjugate hyperbola of $\frac{x^2}{16}-\frac{y^2}{9}=1$ is .

1. $\frac{5}{4}$
2. $\frac{25}{16}$
3. $\frac{5}{3}$
4. $\frac{25}{9}$

Q. A cylinder of the greatest volume is inscribed in a sphere. How many times is the volume of the sphere greater than that of the cylinder?

Q. $P$ and $Q$ are the foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $B$ is an end of the minor axis. If $△PBQ$ is an equilateral triangle, the eccentricity of the ellipse is:

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

Q. If e and e' be the eccentricities of a hyperbola and its conjugate then
$\frac{1}{e^2}+\frac{1}{(e^{\prime }{)}^2}=1$.

1. True
2. False