The asymptotes of the hyperbola frac{x^2}{a^2 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The equation of the hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. The asymptotes of the hyperbola are given by $\frac{x}{a} \pm \frac{y}{b} =

Vedclass · Accepted Answer

$\sec (\alpha)$

Vedclass · Answer

(A) The equation of the hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. The asymptotes of the hyperbola are given by $\frac{x}{a} \pm \frac{y}{b} = 0$. It is a standard property that the area of the triangle formed by any tangent to the hyperbola and its asymptotes is constant and equal to $ab$. Given that the area is $a^2 	an(\alpha)$,we have $ab = a^2 	an(\alpha)$,which implies $b = a 	an(\alpha)$. The eccentricity $e$ of the hyperbola is given by $e = \sqrt{1 + \frac{b^2}{a^2}}$. Substituting $b = a 	an(\alpha)$,we get $e = \sqrt{1 + \frac{a^2 	an^2(\alpha)}{a^2}} = \sqrt{1 + 	an^2(\alpha)} = \sqrt{\sec^2(\alpha)} = \sec(\alpha)$.

The asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,with any tangent to the hyperbola form a triangle whose area is $a^2 \tan (\alpha)$. Then its eccentricity equals

Similar Questions

The product of the perpendicular distances drawn from any point on the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ to its asymptotes is

$A$ hyperbola has its centre at the origin,passes through the point $(4, 2)$ and has a transverse axis of length $4$ along the $x$-axis. Then the eccentricity of the hyperbola is

If $P(\theta) = (x_1, \frac{3 \sqrt{5}}{2})$,$0 < \theta < \frac{\pi}{2}$ is a point on the hyperbola $\frac{x^2}{25} - \frac{y^2}{9} = 1$,where $\theta$ is the parameter in its parametric form,then $2 x_1 + 9 \sin^2 \theta = $

The eccentricity of the hyperbola $4x^2 - 9y^2 = 36$ is

If the length of the transverse and conjugate axes of a hyperbola are $8$ and $6$ respectively,then the difference of the focal distances of any point on the hyperbola will be

Vedclass Test Series

Exam Paper Generator

Online Exam Module