Question

Let the latus rectum subtends a right angle at the center of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If $e$ is the eccentricity of the hyperbola then $\left[e\right]$ is
where $[.]$ represents the greatest integer function.

Answer 1

Coordinates of the end points of the latus rectum of the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $A(ae,\frac{b^2}{a})\text{and}B(ae,-\frac{b^2}{a})$
Let $O$ be the origin
$\therefore O{A}^2+O{B}^2=A{B}^2\Rightarrow 2(ae{)}^2+2\frac{b^4}{a^2}=\frac{4b^4}{a^2}e=\frac{b^2}{a^2}\cdots (\because e^2=1+\frac{b^2}{a^2})e^2-1=e\Rightarrow e=\frac{\sqrt{5}+1}{2}\Rightarrow 1<e<2\Rrightarrow [e]=1$