Question

Let $e_1$ and $e_2$ be the eccentricities of the ellipse $\frac{x^2}{25}+\frac{y^2}{b^2}=1 \left(b<5\right)$ and the hyperbola $\frac{x^2}{16}-\frac{y^2}{b^2}=1$ respectively satisfying ${\mathrm{e}}_1{\mathrm{e}}_2=1$. If $\alpha$ and $\beta$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $\left(\alpha , \beta \right)$ is equal to:

JEE Main 2020 (03 Sep Shift 2)

Options

• A: $\left(8, 10\right)$
• B: $\left(\frac{20}{3}, 12\right)$
• C: $\left(8, 12\right)$
• D: $\left(\frac{24}{5}, 10\right)$

Question

Let $a>0, b>0$. Let $e$ and $l$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. Let $e^{\text{'}}$ and $l^{\text{'}}$ respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If ${\mathrm{e}}^2=\frac{11}{14}l$ and ${\left(e^{\text{'}}\right)}^2=\frac{11}{8}{l}^{\text{'}}$, then the value of $77a+44b$ is equal to

JEE Main 2022 (28 Jun Shift 2)

Options

• A: $100$
• B: $110$
• C: $120$
• D: $130$

Question

An ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through the vertices of the hyperbola $H:\frac{x^2}{49}-\frac{y^2}{64}=-1$. Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$. Let the product of the eccentricities of $E$ and $H$ be $\frac{1}{2}$. If $l$ is the length of the latus rectum of the ellipse $E$, then the value of $113l$ is equal to _______.

JEE Main 2022 (27 Jul Shift 1)

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Question

Let the hyperbola $H:\frac{x^2}{a^2}-y^2=1$ and the ellipse $E:3x^2+4y^2=12$ be such that the length of latus rectum of $H$ is equal to the length of latus rectum of $E$. If $e_H$ and $e_E$ are the eccentricities of $H$ and $E$ respectively, then the value of $12\left(e_H^2+e_E^2\right)$ is equal to _____.

JEE Main 2022 (24 Jun Shift 2)

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Question

Let $H:\frac{x^2}{a^2}-\frac{y^2}{{ b}^2}=1,a>0, b>0$, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4\left(2\sqrt{2}+\sqrt{14}\right)$. If the eccentricity $H$ is $\frac{\sqrt{11}}{2}$, then value of $a^2+b^2$ is equal to ______.

JEE Main 2022 (29 Jun Shift 1)

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Question

Let $H_n:\frac{x^2}{1+n}-\frac{y^2}{3+n}=1, n\in \mathrm{\mathbb{N}}$. Let $k$ be the smallest even value of $n$ such that the eccentricity of $H_k$ is a rational number. If $l$ is the length of the latus rectum of $H_k$, then $21l$ is equal to

JEE Main 2023 (11 Apr Shift 1)

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Question

Consider a hyperbola $\mathrm{H}$ having centre at the origin and foci on the $\mathrm{x}$-axis. Let $\mathrm{C}_1$ be the circle touching the hyperbola $\mathrm{H}$ and having the centre at the origin. Let $\mathrm{C}_2$ be the circle touching the hyperbola $\mathrm{H}$ at its vertex and having the centre at one of its foci. If areas (in sq units) of $C_1$ and $C_2$ are $36 \pi$ and $4 \pi$, respectively, then the length (in units) of latus rectum of $\mathrm{H}$ is

JEE Main 2024 (04 Apr Shift 2)

Options

• A: $\frac{14}{3}$
• B: $\frac{28}{3}$
• C: $\frac{11}{3}$
• D: $\frac{10}{3}$

Question

Let ${\mathrm{e}}_1$ be the eccentricity of the hyperbola $\frac{{\mathrm{x}}^2}{16}-\frac{{\mathrm{y}}^2}{9}=1$ and ${\mathrm{e}}_2$ be the eccentricity of the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1, \mathrm{a}>\mathrm{b}$, which passes through the foci of the hyperbola. If ${\mathrm{e}}_1{\mathrm{e}}_2=1$, then the length of the chord of the ellipse parallel to the $\mathrm{x}$-axis and passing through $(0,2)$ is :

JEE Main 2024 (27 Jan Shift 2)

Options

• A: $4\sqrt{5}$
• B: $\frac{8\sqrt{5}}{3}$
• C: $\frac{10\sqrt{5}}{3}$
• D: $3\sqrt{5}$

Question

Let $H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4 \sqrt{3}$. Suppose the point $(\alpha, 6), \alpha>0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha, 6)$, then $\alpha^2+\beta$ is equal to

JEE Main 2024 (08 Apr Shift 1)

Options

• A: 172
• B: 171
• C: 169
• D: 170

Question

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3 \alpha^2+2 \beta^2$ is equal to

JEE Main 2024 (09 Apr Shift 2)

Options

• A: 237
• B: 242
• C: 205
• D: 225

Question

If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2}, \frac{14}{3}\sqrt{\frac{2}{5}}\right)$ on the hyperbola, is equal to

JEE Main 2024 (31 Jan Shift 1)

Options

• A: $7\sqrt{\frac{2}{5}}-\frac{8}{3}$
• B: $14\sqrt{\frac{2}{5}}-\frac{4}{3}$
• C: $14\sqrt{\frac{2}{5}}-\frac{16}{3}$
• D: $7\sqrt{\frac{2}{5}}+\frac{8}{3}$