Q. 58 - Calculus

Question

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

Step-by-Step Solution

Answer

Hence, proved 

Step 1: Given information

The given equation of a parabola is$\frac{y^2}{B^2}-\frac{x^2}{A^2} =1$

Stp 2: Graph the parabola and find the vertices and foci.

The graph of the parabola $\frac{y^2}{B^2}-\frac{x^2}{A^2} =1$is ,

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

As, from the graph it is clear that the points with coordinates  $\left(0, \pm \sqrt{A^2+B^2}\right)$

are the foci of the hyperbola.