   Chapter 10.1, Problem 82E

Chapter
Section
Textbook Problem

Hyperbola Consider a hyperbola centered at the origin with a horizontal transverse axis. Use the definition of a hyperbola to derive its standard form x 2 a 2 − y 2 b 2 = 1

To determine

To Prove: The standard equation of a hyperbola using the definition.

Explanation

Given: A hyperbola is a set of points such that their distance between the foci is constant.

Formula Used:

Proof:

Consider P(x, y) be any point on a hyperbola.

Let (a, 0) is the vertex and the points (c, 0) and(-c, 0) represents the foci of hyperbola.

Now sum of the distances from foci to the vertex is (c+a) + (a - c) = 2a, which is shown in below figure:

Now according to the definition:

Distance between two foci is constant that means the length of the major axis is 2a.

Therefore:

(x(c))2+(y0)2(xc)2+(y0)2=2a

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

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

Take square both sides:

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

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