Questions
The
eccentricity of the hyperbola whose asymptotes are $3x + 4y = 2{\text{ and }}4x - 3y + 5 = 0$
- 1
- 2
- $\sqrt 2 $
- $\sqrt 3 $
The equation of the conjugate axis of the hyperbola $\frac{{{{\left( {y - 2} \right)}^2}}}{9} - \frac{{{{\left( {x + 3} \right)}^2}}}{{16}} = 1$ is
- $y=2$
- $y=6$
- $y=8$
- $y=3$
The eccentricity of the conjugate hyperbola of the hyperbola $x^{2} - 3y^{2} = 1$ is
- $\dfrac {2}{\sqrt {3}}$
- $\sqrt {3}$
- $2\sqrt {3}$
- $2$
If variable has its interceptson the coordinates axes $e$ and $e'$ where $e/2$ and $e'/2$ are the eccentricities of hyperbola and conjugate hyperbola, Then the line always touches the circle $x^{2}+y^{2}=r^{2}$, where $r=$
- $1$
- $2$
- $3$
- $Cannot\ be\ decided$
Let $e$ be the eccentricity of a hyperbola $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$, and $f(e)$ be the eccentricity of hyperbola $-\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$, then $\displaystyle \int _{ 1 }^{ 3 } \underbrace { fff.....f\left( e \right) } _{ n\quad times } de$ is equal to
- $2$, if $n$ is even
- $4$, if $n$ is even
- $2\sqrt{2}$, if $n$ is odd
- $4\sqrt{2}$, if $n$ odd
$e _{1}$ and $e _{2}$ are respectively the eccentricities of a hyperbola and its conjugate then $\dfrac{1}{e^{2} _{1}}$+$\dfrac{1}{e^{2} _{2}}$=1.
- True
- False
The eccentricity of the conjugate hyperbola of the hyperbola $x^{2}-3y^{2}=1$ is
- $2$
- $2/\sqrt {3}$
- $4$
- $4/3$
The area of quadrilateral formed by focil hyperbola $\dfrac{x^2}{4}-\dfrac{y^2}{3}=1$ & its conjugate hyperbola is
- $14$
- $24$
- $12$
- $10$
The eccentricity of the hyperbola length of whose conjugate axis is equal to half of the distance betweet the foci is
- $\dfrac{4}{\sqrt{3}}$
- $\dfrac{4}{3}$
- $\dfrac{2}{\sqrt{3}}$
- $\sqrt{3}$
Assertion(A): lf the lines $3x+y+p=0$ and $2x+5y-3=0$ are conjugate with respect to $3x^{2}-2y^{2}=6$ then $\mathrm{p}=1$
Reason(R): lf the lines $l _{1}x+m _{1}y+n _{1}=0$ and $l _{2}x+m _{2}y+n _{2}=0$ are conjugate with respect to the hyperbola $\mathrm{S}=0$ is $a^{2}l _{1}l _{2}+b^{2}m _{1}m _{2}=n _{1}n _{2}$
- Both A and R are true and R is the correct
explanation of A. - Both A and R are true but R is not correct
explanation of A. - A is true but R is false
- A is false but R is true
The equation to the conjugate hyperbola of $2x^{2}-3y^{2}-4x+6y-15=0$ is
- $2x^{2}-3y^{2}-4x+6y+13=0$
- $2x^{2}-3y^{2}-4x+6y-1=0$
- $2x^{2}-3y^{2}-4x+6y+15=0$
- $2x^{2}-3y^{2}-4x+6y-8=0$
If the hyperbolas, $ x^2+3xy+2y^2+2x+3y+2=0 $ and $ x^2+3xy+2y^2+2x+3y+c=0 $ are conjugate of each other, the value of $c$ is equal to
- $-2$
- $4$
- $0$
- $1$
If the line $lx+my+n=0$ meets the hyperbola $\displaystyle \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ at the extermities of a pair of conjugate diameters, then
- $a^{2}l^{2}-b^{2}m^{2}=0$
- $a^{2}l^{2}-b^{2}m^{2}=1$
- $a^{2}l^{2}-b^{2}m^{2}=2$
- $a^{2}l^{2}-b^{2}m^{2}=3$
Find the equation to the hyperbola,conjugate to the hyperbola $ 2x^2+3xy-2y^2-5x+5y+2=0 $.
- $ 2x^2+3xy-2y^2-5x+5y-8=0 $
- $ x^2+3xy-y^2-5x+5y-8=0 $
- $ x^2+3xy-y^2-5x+5y+8=0 $
- None of these
The equation of a hyperbola, conjugate to the hyperbola $x^2+3xy+2y^2+2x+3y=0$ is?
- $x^2+3xy+2y^2+2x+3y+1=0$
- $x^2+3xy+2y^2+2x+3y+2=0$
- $x^2+3xy+2y^2+2x+3y+3=0$
- $x^2+3xy+2y^2+2x+3y+4=0$