Let eccentricity of hyperbola H : frac{ x ^{2}}{ a ^{2}}-frac{ y ^{2}}{ b ^{2}}=1 be √{frac{5}{2}}

English

Gujarati

Hindi

10-2. Parabola, Ellipse, Hyperbola

hard

Let the eccentricity of the hyperbola $H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$ be $\sqrt{\frac{5}{2}}$ and length of its latus rectum be $6 \sqrt{2}$, If $y =2 x + c$ is a tangent to the hyperbola $H$, then the value of $c ^{2}$ is equal to

$18$

$20$

$24$

$32$

(JEE MAIN-2022)

Solution

$y = mx \pm \sqrt{ a ^{2} m ^{2}- b ^{2}}$

$m =2, c ^{2}= a ^{2} m ^{2}- b ^{2}$

$c ^{2}=4 a ^{2}- b ^{2}$

$e ^{2}=1+\frac{ b ^{2}}{ a ^{2}}$

$\frac{5}{2}=1+\frac{b^{2}}{a^{2}}$

$\frac{3}{2}=\frac{b^{2}}{a^{2}} \Rightarrow b^{2}=\frac{3 a^{2}}{2}$

$\frac{2 b^{2}}{a}=6 \sqrt{2}$

$\frac{2}{a} \times \frac{3 a^{2}}{2}=6 \sqrt{2}$

$3 a=6 \sqrt{2}$

$a=2 \sqrt{2}$

$b^{2}=\frac{3}{2} \times 8=12$

$b=2 \sqrt{3}$

$\therefore c^{2}=4 \times 8-12$

$c^{2}=20$

Standard 11

Mathematics

Let the latus rectum of the hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtend an angle of $\frac{\pi}{3}$ at the centre of the hyperbola. If $\mathrm{b}^2$ is equal to $\frac{l}{\mathrm{~m}}(1+\sqrt{\mathrm{n}})$, where $l$ and $\mathrm{m}$ are co-prime numbers, then $l^2+\mathrm{m}^2+\mathrm{n}^2$ is equal to______________.

hard

(JEE MAIN-2024)

The locus of the centroid of the triangle formed by any point $\mathrm{P}$ on the hyperbola $16 \mathrm{x}^{2}-9 \mathrm{y}^{2}+$ $32 x+36 y-164=0$, and its foci is:

hard

(JEE MAIN-2021)

A common tangent $T$ to the curves $C_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ and $C_{2}: \frac{x^{2}}{42}-\frac{y^{2}}{143}=1$ does not pass through the fourth quadrant. If $T$ touches $C _{1}$ at ( $\left.x _{1}, y _{1}\right)$ and $C _{2}$ at $\left( x _{2}, y _{2}\right)$, then $\left|2 x _{1}+ x _{2}\right|$ is equal to $……$

normal

(JEE MAIN-2022)

The equation of the director circle of the hyperbola $\frac{{{x^2}}}{{16}} – \frac{{{y^2}}}{4} = 1$ is given by

easy

If $5x + 9 = 0$ is the directrix of the hyperbola $16x^2 -9y^2 = 144,$ then its corresponding focus is

hard

(JEE MAIN-2019)

Let the eccentricity of the hyperbola $H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$ be $\sqrt{\frac{5}{2}}$ and length of its latus rectum be $6 \sqrt{2}$, If $y =2 x + c$ is a tangent to the hyperbola $H$, then the value of $c ^{2}$ is equal to

Solution

Similar Questions

The locus of the centroid of the triangle formed by any point $\mathrm{P}$ on the hyperbola $16 \mathrm{x}^{2}-9 \mathrm{y}^{2}+$ $32 x+36 y-164=0$, and its foci is:

The equation of the director circle of the hyperbola $\frac{{{x^2}}}{{16}} – \frac{{{y^2}}}{4} = 1$ is given by

If $5x + 9 = 0$ is the directrix of the hyperbola $16x^2 -9y^2 = 144,$ then its corresponding focus is

Start a Free Trial Now