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Chapter 10: Problem 77

Convert the polar equation to rectangular form. $$r=\frac{1}{1-\cos \theta}$$

Short Answer

Expert verified
The particular exercise has been converted to the rectangular form as required using the relationships between polar and rectangular coordinates.

Step by step solution

01

Substitute for cos theta

Our starting equation is \(r = 1/(1 - \cos\theta)\). We now use the formula for cos theta in polar coordinates. Thus, replace \(\cos\theta\) by \(x/r\) Getting: \(r = 1/(1 - x/r)\)
02

Simplify the equation

Clear the denominator by multiplying the entire equation by \((1 - x/r)\), then by r\n\(r^2 = r - x\)
03

Substitute for r squared

Now, we utilize the formula \(r^2 = x^2 + y^2\) from polar to rectangular conversion. Thus, replace \(r^2\) by \(x^2 + y^2\) Resulting in: \(x^2 + y^2 = r - x\)
04

Final Simplification

We need to get rid of r in the equation. So we replace r with its equivalent in the rectangular form r = sqrt(x^2 + y^2). Substituting this into our equation we obtain the following: \(x^2 + y^2 = \sqrt{x^2 + y^2} - x\), which can now be solved for x and y

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Most popular questions from this chapter

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