   Chapter 10.6, Problem 70E

Chapter
Section
Textbook Problem

# Eccentricity In Exercises 67 and 68, let r 0 represent the distance from a focus to the nearest vertex, and let r 1 represent the distance from the focus to the farthest vertex.Show that the eccentricity of a hyperbola can be written as e = r 1 + r 0 r 1 − r 0 . Then show that r 1 r 0 = e + 1 e − 1 .

To determine

To prove:

The eccentricity of hyperbola can be written as e=r1+r0r1r0 then show that r`1r0=e+1e1.

Explanation

Given:

r0 represents the distance from the focus to nearest vertex.

r1 represents the distance from the focus to the farthest vertex.

Formula Used:

a=12(r1r0)

e=ca

Proof:

As r0 represents the distance from the focus to nearest vertex and r1 represents the distance from the focus to the farthest vertex.

Then for hyperbola,

r1=c+a and r0=ca.

Now subtract r1 with r0.

Then, r1r0=2a.

Similarly,

Then, r1+r0=2c.

Now e=ca,

ca=r1+r0r1r0

Use componendo and dividendotheorems to the above equation

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