Math Problem Statement

Convert the equation x + 3y + 6 = 0 to polar form. Then solve the resulting equation for r.

Solution

To convert the equation x+3y+6=0x + 3y + 6 = 0 into polar form, we use the relationships between Cartesian coordinates (x,y)(x, y) and polar coordinates (r,θ)(r, \theta), which are:

x=rcosθandy=rsinθx = r \cos \theta \quad \text{and} \quad y = r \sin \theta

Step-by-Step Conversion:

  1. Start with the original equation: x+3y+6=0x + 3y + 6 = 0

  2. Substitute x=rcosθx = r \cos \theta and y=rsinθy = r \sin \theta: rcosθ+3rsinθ+6=0r \cos \theta + 3r \sin \theta + 6 = 0

  3. Factor out rr: r(cosθ+3sinθ)=6r(\cos \theta + 3 \sin \theta) = -6

  4. Solve for rr: r=6cosθ+3sinθr = \frac{-6}{\cos \theta + 3 \sin \theta}

Thus, the polar form of the equation is:

r=6cosθ+3sinθr = \frac{-6}{\cos \theta + 3 \sin \theta}

Would you like further explanation on how the polar form was derived, or do you have any other questions?

Related Questions:

  1. How do you convert any general Cartesian equation to polar form?
  2. What is the significance of the polar coordinate system in mathematics?
  3. How does the polar coordinate system simplify equations of circles or lines?
  4. Can you show how polar and Cartesian coordinates relate geometrically?
  5. What are common applications of polar coordinates in physics?

Tip: In polar coordinates, the angle θ\theta represents the direction of the point from the origin, while rr represents the distance from the origin. This system is useful when analyzing problems with rotational symmetry.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Polar Coordinates

Formulas

x = r * cos(θ)
y = r * sin(θ)
r = -6 / (cos(θ) + 3sin(θ))

Theorems

Conversion from Cartesian to Polar Coordinates

Suitable Grade Level

Grades 10-12