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Chapter 6: Problem 61

Convert the polar equation to rectangular form. $$r=\frac{2}{1+\sin \theta}$$

Short Answer

Expert verified
The rectangular format of the equation \( r = \frac{2}{1+ \sin \theta} \) is \( x^2 + 2y^2 + 2y - 4 = 0 \)

Step by step solution

01

Identify the Polar Equation

The given polar equation is \( r= \frac{2}{1+ \sin \theta} \).
02

Multiplication to Clear Fraction

Multiply both sides of the equation by the denominator of the right hand side to get rid of the fraction: \( r(1+ \sin \theta) = 2 \).
03

Convert Equation

In this step, replace \( r \) with \( \sqrt{x^2 + y^2} \) and \( \sin \theta \) with \( \frac{y}{r} \), obtaining \( \sqrt{x^2 + y^2}(1 + \frac{y}{\sqrt{x^2 + y^2}}) = 2 \)
04

Simplify Equation

Simplify the equation to form \( \sqrt{x^2 + y^2} + y = 2 \)
05

Final Rectangular Form

Square both sides to change the form to a rectangular equation: \( (x^2 + y^2 + 2y + y^2) - 4 = 0 \), which simplifies to \( x^2 + 2y^2 + 2y - 4 = 0 \)

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Most popular questions from this chapter

In Exercises 73-78, solve the trigonometric equation. $$ \sqrt{2} \sec \theta=2 \csc \frac{\pi}{4} $$

\(r=4+3 \cos \theta\)

In Exercises 25-28, use a graphing utility to graph the polar equation. Identify the graph. $$ r=\frac{-5}{2+4 \sin \theta} $$

In Exercises 29-32, use a graphing utility to graph the rotated conic. $$ r=\frac{2}{1-\cos (\theta-\pi / 4)} $$

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{lll} {\text { Conic }} & \text { Eccentricity } & \text { Directrix } \\ \text { Parabola } & e=1 & x=-1 \\ \end{array} $$
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