To convert the given Cartesian equation, x^2 + y^2 = 4x, to a polar equation, we need to express x and y in terms of polar coordinates. In polar coordinates, a point is represented by its distance from the origin (r) and the angle it makes with the positive x-axis (θ).
To begin, let's express x and y in terms of r and θ using the following relationships:
x = r * cos(θ)
y = r * sin(θ)
Substituting these expressions into the given Cartesian equation, we have:
(r * cos(θ))^2 + (r * sin(θ))^2 = 4 * (r * cos(θ))
Expanding and simplifying this equation, we get:
r^2 * cos^2(θ) + r^2 * sin^2(θ) = 4r * cos(θ)
Using the trigonometric identity cos^2(θ) + sin^2(θ) = 1, we can simplify further:
r^2 + r^2 * sin^2(θ) = 4r * cos(θ)
Factoring out an r from the left-hand side of the equation, we obtain:
r^2 * (1 + sin^2(θ)) = 4r * cos(θ)
Dividing both sides of the equation by r, we get:
r * (1 + sin^2(θ)) = 4 * cos(θ)
Finally, rearranging the terms, we arrive at the polar equation for the given Cartesian equation:
r = 4 * cos(θ) / (1 + sin^2(θ))
In polar form, this equation represents a curve in the polar coordinate system.