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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 10: Q 34. (page 683)

Convert each polar equation into a rectangular equation.
$r (2 - \cos (胃)) = 2$

Short Answer

Expert verified

The rectangular equation is

$3 x^{2} + 4 y^{2} - 4 x = 4$

Step by step solution

01

Step 1. Given information

The polar equation is

$r (2 - \cos (胃)) = 2$

02

Step 2. Convert into the rectangular equation

The polar equation is

$r (2 - \cos (胃)) = 2$

Multiply left side term by $r$

$2 r - r \cos (胃) = 2$

Apply the conversion formula

$r = \sqrt{x^{2} + y^{2}}, x = r \cos (胃)$

$2 \sqrt{x^{2} + y^{2}} - x = 2$

Add both sides by $x$

$2 \sqrt{x^{2} + y^{2}} - x + x = 2 + x 2 \sqrt{x^{2} + y^{2}} = 2 + x$

Take square on both sides and simplify

${(2 \sqrt{x^{2} + y^{2}})}^{2} = {(2 + x)}^{2} 4 (x^{2} + y^{2}) = 4 + x^{2} + 4 x 4 x^{2} + 4 y^{2} = 4 + x^{2} + 4 x 4 x^{2} + 4 y^{2} - x^{2} - 4 x = 4 3 x^{2} + 4 y^{2} - 4 x = 4$

So, the rectangular equation is

$3 x^{2} + 4 y^{2} - 4 x = 4$

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Most popular questions from this chapter

In Problems 43鈥�52, identify the graph of each equation without applying a rotation of axes.

2 x^{2} - 3 x y + 2 y^{2} - 4 x - 2 = 0

Find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility.

$y^{2} = - 16 x$

In Problems 31鈥� 42, rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation. Refer to Problems 21鈥�30 for Problems 31鈥� 40.

$3 x^{2} - 10 x y + 3 y^{2} - 32 = 0$

In Problems 15鈥�18, the graph of a hyperbola is given. Match each graph to its equation.

Equations: $(A) \frac{x^{2}}{4} - y^{2} = 1 (B) x^{2} - \frac{y^{2}}{4} = 1 (C) \frac{y^{2}}{4} - x^{2} = 1 (D) y^{2} - \frac{x^{2}}{4} = 1$

Graph:

Graph each function.

$f (x) = \sqrt{16 - 4 x^{2}}$ .

See all solutions

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