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(x+7)^2+y^2=49 In Polar Form

2 min read Jun 17, 2024
(x+7)^2+y^2=49 In Polar Form

Converting (x+7)^2 + y^2 = 49 to Polar Form

The equation (x+7)^2 + y^2 = 49 represents a circle with center (-7, 0) and radius 7. Let's convert this equation into polar form (r, θ).

Recall the Conversion Formulas

  • x = r cos(θ)
  • y = r sin(θ)

Substitution and Simplification

  1. Substitute: Substitute the polar form expressions for x and y into the given equation: (r cos(θ) + 7)^2 + (r sin(θ))^2 = 49

  2. Expand: Expand the equation: r^2 cos^2(θ) + 14r cos(θ) + 49 + r^2 sin^2(θ) = 49

  3. Simplify: Combine the terms with r^2 and simplify: r^2 (cos^2(θ) + sin^2(θ)) + 14r cos(θ) = 0

  4. Trigonometric Identity: Utilize the identity cos^2(θ) + sin^2(θ) = 1: r^2 + 14r cos(θ) = 0

  5. Factor: Factor out r: r(r + 14 cos(θ)) = 0

  6. Solve for r:

    • r = 0
    • r + 14 cos(θ) = 0 => r = -14 cos(θ)

Final Polar Form

The polar form of the equation (x+7)^2 + y^2 = 49 is:

r = -14 cos(θ)

Notice that the solution r = 0 corresponds to the origin, which is a point on the circle. The solution r = -14 cos(θ) represents the entire circle, excluding the origin. This is because the equation defines a circle centered at (-7, 0), which is shifted 7 units to the left of the origin.

This polar form provides a more compact and intuitive way to represent the circle, especially when dealing with rotations and other transformations.

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