(x-1)^2+y^2=1 In Polar Coordinates

Jasonbradley

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Jun 17, 2024

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Transforming (x-1)^2 + y^2 = 1 to Polar Coordinates

The equation (x-1)^2 + y^2 = 1 represents a circle with a radius of 1 and centered at (1,0) in the Cartesian coordinate system. Let's explore how to transform this equation into polar coordinates.

From Cartesian to Polar Coordinates

We use the following relationships:

• x = r cos(θ)
• y = r sin(θ)

Substituting these into the original equation:

(r cos(θ) - 1)^2 + (r sin(θ))^2 = 1

Expanding and simplifying:

r^2 cos^2(θ) - 2r cos(θ) + 1 + r^2 sin^2(θ) = 1

r^2 (cos^2(θ) + sin^2(θ)) - 2r cos(θ) = 0

Using the trigonometric identity cos^2(θ) + sin^2(θ) = 1:

r^2 - 2r cos(θ) = 0

Solving for r

To express the equation in terms of r, we factor out r:

r(r - 2cos(θ)) = 0

This leads to two possible solutions:

• r = 0 This solution represents the origin.
• r = 2cos(θ) This solution represents the circle with a radius of 1 centered at (1,0).

Interpretation in Polar Coordinates

The equation r = 2cos(θ) describes the circle in polar coordinates. This equation tells us that for each angle θ, the radius r is determined by 2 times the cosine of θ.

This makes sense when we consider the behavior of the equation:

• When θ = 0: r = 2, which corresponds to the point (2,0) on the rightmost point of the circle.
• When θ = π/2: r = 0, which corresponds to the point (0,0) at the origin.
• When θ = π: r = -2, which corresponds to the point (-2, π) on the leftmost point of the circle (remember that negative r values mean we move in the opposite direction of θ).

Therefore, the equation r = 2cos(θ) accurately describes the circle (x-1)^2 + y^2 = 1 in polar coordinates.