To convert the Cartesian equation x² + y² = 2cx into its polar form, we start by recalling the relationships between Cartesian and polar coordinates. In polar coordinates, x and y are expressed as:
x = r * cos(θ)
y = r * sin(θ)
Where r is the radial distance from the origin and θ is the angle from the positive x-axis. We will substitute these expressions into the given equation.
Now, substituting for x and y, we have:
(r * cos(θ))² + (r * sin(θ))² = 2c(r * cos(θ))
Expanding the left side, we get:
r² * cos²(θ) + r² * sin²(θ) = 2c * r * cos(θ)
By employing the Pythagorean identity cos²(θ) + sin²(θ) = 1, the left side simplifies to:
r² = 2c * r * cos(θ)
To isolate r, we can rearrange this equation:
r² – 2c * r * cos(θ) = 0
This is a quadratic equation in terms of r. We can apply the quadratic formula, r = (-b ± √(b² – 4ac)) / 2a, where:
- a = 1
- b = -2c * cos(θ)
- c = 0
Substituting these values into the quadratic formula gives:
r = (2c * cos(θ) ± √((2c * cos(θ))² – 4 * 1 * 0)) / (2 * 1)
Which simplifies to:
r = c * cos(θ) ± √(c² * cos²(θ))
Since we are only interested in the positive solution for r, we have:
r = c * cos(θ) + |c * cos(θ)|
This yields:
r = 2c * cos(θ)
Therefore, the polar equation corresponding to the Cartesian equation x² + y² = 2cx is:
r = 2c * cos(θ)
This polar equation represents a circle with radius c centered at (c, 0) in the Cartesian coordinate system.