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.