Let $e_1$ be the eccentricity of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $e_2$ be the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$, which passes through the foci of the hyperbola. If $e_1 e_2=1$, then the length of the chord of the ellipse parallel to the $\mathrm{x}$-axis and passing through $(0,2)$ is :
Let $e_1$ be the eccentricity of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $e_2$ be the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$, which passes through the foci of the hyperbola. If $e_1 e_2=1$, then the length of the chord of the ellipse parallel to the $\mathrm{x}$-axis and passing through $(0,2)$ is :
- [JEE MAIN 2024]
- A
$4 \sqrt{5}$
- B
$\frac{8 \sqrt{5}}{3}$
- C
$\frac{10 \sqrt{5}}{3}$
- D
$3 \sqrt{5}$
Similar Questions
Area of the quadrilateral formed with the foci of the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ and $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = - 1$ is
Area of the quadrilateral formed with the foci of the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ and $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = - 1$ 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:
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]
The equation of the director circle of the hyperbola $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{4} = 1$ is given by
The equation of the director circle of the hyperbola $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{4} = 1$ is given by
Find the equation of the hyperbola satisfying the give conditions: Foci $(\pm 3 \sqrt{5},\,0),$ the latus rectum is of length $8$
Find the equation of the hyperbola satisfying the give conditions: Foci $(\pm 3 \sqrt{5},\,0),$ the latus rectum is of length $8$
The equation of the transverse and conjugate axis of the hyperbola $16{x^2} - {y^2} + 64x + 4y + 44 = 0$ are
The equation of the transverse and conjugate axis of the hyperbola $16{x^2} - {y^2} + 64x + 4y + 44 = 0$ are