   Chapter 10.5, Problem 52E

Chapter
Section
Textbook Problem

# Use the definition of a hyperbola to derive Equation 6 for a hyperbola with foci (±c, 0) and vertices (±a, 0).

To determine

To Find: The derivation of Equation 6 for a hyperbola with foci (±c,0) and vertices (±a,0) .

Explanation

Given:

The foci and vertices are (±c,0) and (±a,0) respectively.

Derivation:

The definition of the hyperbola as below,

|PF1||PF2|=±2a

Substitute the value of (x,y) for P , (c,0) for F1 and (c,0) for F2 .

(x+c)2+y2(xc)2+y2=±2a(x+c)2+y2=(xc)2+y2±2a

Take the square on both sides.

((x+c)2+y2)2=((xc)2+y2±2a)2(x+c)2+y2=(xc)2+y2+4a2±4a(xc)2+y2(x+c)2+y2=(xc)2+y2+4a2±4a(xc)2+y2

(x+c)2(xc)24a2=±4a(xc)2+y2(x+cx+c)(x+c+xc)4a2=±4a(xc)2+y2(2c)(2x)4a2=±4a(xc)2+y2 .

(2c)(2x)4a2=±4a(xc)2+y24cx4a2=±4a(xc)2+y2cxa2=±4a(xc)2+y2

Take square on both sides

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