a circle is formed with a center (-3,5) and a radius.of 4 write the circle as a pair of parametric equations. then show how you can eliminate the parameter to return to the rectangular form
x = -3 + 4cosθ
y = 5 + 4sinθ
well, we all know that the x-y equation is
(x+3)^2 + (y-5)^2 = 16
To write the circle as a pair of parametric equations, we need to express the x and y coordinates in terms of a parameter. Let's use the parameter θ.
First, let's define the center of the circle as (h, k). In this case, the center is (-3, 5).
x = h + r * cos(θ)
y = k + r * sin(θ)
Here, r represents the radius, and θ is the parameter angle. Since the radius of our circle is 4, we have r = 4.
Substituting the values of the center and the radius into the equations, we get:
x = -3 + 4 * cos(θ)
y = 5 + 4 * sin(θ)
These are the parametric equations for the circle with a center (-3, 5) and a radius of 4.
To eliminate the parameter (θ) and return to the rectangular form, you can use the following steps:
1. Start with the parametric equations: x = -3 + 4 * cos(θ) and y = 5 + 4 * sin(θ).
2. Square both equations: x^2 = (-3 + 4 * cos(θ))^2 and y^2 = (5 + 4 * sin(θ))^2.
3. Rearrange the first equation: (x + 3)^2 = (4 * cos(θ))^2.
4. Rearrange the second equation: (y - 5)^2 = (4 * sin(θ))^2.
5. Expand and simplify both equations:
(x + 3)^2 = (16 * cos^2(θ))
(y - 5)^2 = (16 * sin^2(θ))
6. Apply the trigonometric identity, sin^2(θ) + cos^2(θ) = 1:
(x + 3)^2 = 16 * (1 - sin^2(θ))
(y - 5)^2 = 16 * sin^2(θ)
7. Simplify further:
(x + 3)^2 = 16 - 16 * sin^2(θ)
(y - 5)^2 = 16 * sin^2(θ)
8. Combine both equations:
(x + 3)^2 + (y - 5)^2 = 16
9. Rearrange the equation to the standard form of a circle:
(x + 3)^2 + (y - 5)^2 = 4^2
This is the rectangular form of the circle equation for the given parameters.