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Chapter 10: Q. 43 (page 683)

Derive equation (b) in Table 5:

r=ep1+ecos鈦�胃

Short Answer

Expert verified

The polar equationr=ep1+ecos鈦�胃is proved.

Step by step solution

01

Step 1. Given information

We have to derive the following equation :

r=ep1+ecos鈦�胃

02

Step 2. Examine the given polar equation

To do that, let us examine first the given polar equation. We can observed that the denominator has a+cos胃term which means that the directrix is perpendicular to the polar axis and is located p units to the right of the pole.

03

Step 3. Find the value of D

Based from the figure,

D=p-l

Find the value of l,

cos鈦�胃=lr鈬�l=rcos鈦�胃

Substitute the value,

D=p-rcos胃

04

Step 4. Derive the equation

Relation between radius and eccentricity is given by :

r=eD

Apply previous equation to the equation of r,

r=e(p鈭�rcos鈦�胃)r=(1)ep鈭�ercos鈦�胃r+ercos鈦�胃=(2)epr(1+ecos鈦�胃)=(3)epr=(4)ep1+ecos鈦�胃

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