Chapter 7.3, Problem LC

(a)

To determine

To calculate: The eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ when its asymptotes are perpendicular.

(b)

To determine

To prove: The relation $e_1^2{e}_2^2=e_1^2+e_2^2$ where $e_1$ and $e_2$ are the eccentricities of the hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$ respectively.

(c)

To determine

To prove: The length of the shorter focal radius is $x_{1}e-a$ and the length of the longer focal radius is $x_{1}e+a$ for a point $\left(x_1,y_1\right)$ on the right-hand branch of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$