Math Problem Statement

Write the given polar equation r = 3 / (1 - cos(θ)) using rectangular coordinates (x, y).

Solution

To convert the given polar equation r=31cosθr = \frac{3}{1 - \cos\theta} into rectangular coordinates (x,yx, y), we will use the relationships between polar and rectangular coordinates:

  1. x=rcosθx = r \cos\theta
  2. y=rsinθy = r \sin\theta
  3. r2=x2+y2r^2 = x^2 + y^2
  4. cosθ=xr\cos\theta = \frac{x}{r}

Step-by-Step Conversion:

  1. Start with the polar equation: r=31cosθr = \frac{3}{1 - \cos\theta}

  2. Multiply through by 1cosθ1 - \cos\theta to eliminate the denominator: r(1cosθ)=3r(1 - \cos\theta) = 3

  3. Expand: rrcosθ=3r - r\cos\theta = 3

  4. Substitute rcosθ=xr\cos\theta = x: rx=3r - x = 3

  5. Solve for rr: r=x+3r = x + 3

  6. Substitute r2=x2+y2r^2 = x^2 + y^2: (x+3)2=x2+y2(x + 3)^2 = x^2 + y^2

  7. Expand (x+3)2(x + 3)^2: x2+6x+9=x2+y2x^2 + 6x + 9 = x^2 + y^2

  8. Simplify by canceling x2x^2 from both sides: 6x+9=y26x + 9 = y^2

  9. Rearrange to get the general form: 0=y26x90 = y^2 - 6x - 9

Final Answer:

The rectangular form of the equation is: 0=y26x90 = y^2 - 6x - 9

Would you like a more detailed explanation of any step or have questions?


Here are 5 related questions for further exploration:

  1. How do we convert other polar equations, such as r=2sinθr = 2\sin\theta, to rectangular form?
  2. What type of conic section does this rectangular equation represent?
  3. Can the equation be rewritten in standard form for a parabola?
  4. How would the equation change if cosθ\cos\theta was replaced with sinθ\sin\theta?
  5. How do the transformations x=rcosθx = r\cos\theta and y=rsinθy = r\sin\theta work geometrically?

Tip: Always check if the final rectangular equation corresponds to a familiar geometric shape (circle, parabola, ellipse, or hyperbola) for a better understanding of the problem!

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Math Problem Analysis

Mathematical Concepts

Polar Coordinates
Rectangular Coordinates
Converting Coordinate Systems
Equations of Conic Sections

Formulas

x = r * cos(θ)
y = r * sin(θ)
r^2 = x^2 + y^2
cos(θ) = x / r

Theorems

Relationship between Polar and Rectangular Coordinates

Suitable Grade Level

Grades 10-12