The eccentricity of the hyperbola whose length of the latus rectum is equal to $8$ and the length of its conjugate axis is equal to half of the distance between its foci is :
The eccentricity of the hyperbola whose length of the latus rectum is equal to $8$ and the length of its conjugate axis is equal to half of the distance between its foci is :
- [JEE MAIN 2016]
- A
$\frac{2}{{\sqrt 3 }}\;$
- B
$\sqrt 3 $
- C
$\frac{4}{3}$
- D
$\frac{4}{{\sqrt 3 }}$
Similar Questions
Let $H : \frac{ x ^2}{ a ^2}-\frac{ y ^2}{ b ^2}=1$, where $a > b >0$, be $a$ hyperbola in the $xy$-plane whose conjugate axis $LM$ subtends an angle of $60^{\circ}$ at one of its vertices $N$. Let the area of the triangle $LMN$ be $4 \sqrt{3}$..
List $I$
List $II$
$P$ The length of the conjugate axis of $H$ is
$1$ $8$
$Q$ The eccentricity of $H$ is
$2$ ${\frac{4}{\sqrt{3}}}$
$R$ The distance between the foci of $H$ is
$3$ ${\frac{2}{\sqrt{3}}}$
$S$ The length of the latus rectum of $H$ is
$4$ $4$
The correct option is:
Let $H : \frac{ x ^2}{ a ^2}-\frac{ y ^2}{ b ^2}=1$, where $a > b >0$, be $a$ hyperbola in the $xy$-plane whose conjugate axis $LM$ subtends an angle of $60^{\circ}$ at one of its vertices $N$. Let the area of the triangle $LMN$ be $4 \sqrt{3}$..
| List $I$ | List $II$ |
| $P$ The length of the conjugate axis of $H$ is | $1$ $8$ |
| $Q$ The eccentricity of $H$ is | $2$ ${\frac{4}{\sqrt{3}}}$ |
| $R$ The distance between the foci of $H$ is | $3$ ${\frac{2}{\sqrt{3}}}$ |
| $S$ The length of the latus rectum of $H$ is | $4$ $4$ |
The correct option is:
- [IIT 2018]
Consider a hyperbola $H : x ^{2}-2 y ^{2}=4$. Let the tangent at a point $P (4, \sqrt{6})$ meet the $x$ -axis at $Q$ and latus rectum at $R \left( x _{1}, y _{1}\right), x _{1}>0 .$ If $F$ is a focus of $H$ which is nearer to the point $P$, then the area of $\Delta QFR$ is equal to ....... .
Consider a hyperbola $H : x ^{2}-2 y ^{2}=4$. Let the tangent at a point $P (4, \sqrt{6})$ meet the $x$ -axis at $Q$ and latus rectum at $R \left( x _{1}, y _{1}\right), x _{1}>0 .$ If $F$ is a focus of $H$ which is nearer to the point $P$, then the area of $\Delta QFR$ is equal to ....... .
- [JEE MAIN 2021]
$P(6, 3)$ is a point on the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ . If the normal at point $P$ intersect the $x-$ axis at $(10, 0)$ , then the eccentricity of the hyperbola is
$P(6, 3)$ is a point on the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ . If the normal at point $P$ intersect the $x-$ axis at $(10, 0)$ , then the eccentricity of the hyperbola is
The equation of the tangent to the hyperbola $4{y^2} = {x^2} - 1$ at the point $(1, 0)$ is
The equation of the tangent to the hyperbola $4{y^2} = {x^2} - 1$ at the point $(1, 0)$ is
Let $P \left( x _0, y _0\right)$ be the point on the hyperbola $3 x ^2-4 y ^2$ $=36$, which is nearest to the line $3 x+2 y=1$. Then $\sqrt{2}\left( y _0- x _0\right)$ is equal to :
Let $P \left( x _0, y _0\right)$ be the point on the hyperbola $3 x ^2-4 y ^2$ $=36$, which is nearest to the line $3 x+2 y=1$. Then $\sqrt{2}\left( y _0- x _0\right)$ is equal to :
- [JEE MAIN 2023]