Question

Prove Theorem 9.20 (a). That is, show that the graph of the equation $\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$satisfies Definition 9.19, where the points with coordinates $(\pm \sqrt{A^2+B^2},0)$are the foci of the hyperbola

Short answer

Hence, proved.

Step-by-step solution

Given information

The given equation of the hyperbola is,

$\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$

Plot the graph of the hyperbola.

The graph of the hyperbola is,

The coordinates of the hyperbola are $(\pm \sqrt{A^2+B^2},0)$.

And, the foci of the parabola is $(\pm \sqrt{A^2+B^2},0)$

Hence preoved.