A circle is formed with a center (-3, 5) and a radius 4 write a pair of parametric equations then show how to return to rectangular form
To write a pair of parametric equations for a circle with a given center and radius, we can use the formulas:
x = x_center + r * cos(t)
y = y_center + r * sin(t)
Where (x, y) are the coordinates of a point on the circle, (x_center, y_center) are the coordinates of the center of the circle, r is the radius, and t is the parameter that varies as we move along the circle.
For the given circle with a center (-3, 5) and a radius of 4, the parametric equations can be written as:
x = -3 + 4 * cos(t)
y = 5 + 4 * sin(t)
To return to the rectangular form (also known as the Cartesian form), we need to eliminate the parameter t from the equations. We can do this by replacing t with the appropriate value from the unit circle.
The unit circle is a circle with a radius of 1, centered at the origin (0, 0). Using the unit circle, we can determine the values of cos(t) and sin(t) for any given angle.
In this case, since we have a circle with a center at (-3, 5) and a radius of 4, we need to shift our reference point on the unit circle.
Let's consider a new angle α such that cos(α) = x_center/r and sin(α) = y_center/r.
Substituting these values into the parametric equations, we get:
x = -3 + 4 * (cos(α) * cos(t) - sin(α) * sin(t))
y = 5 + 4 * (sin(α) * cos(t) + cos(α) * sin(t))
To simplify further, we can use the trigonometric identity:
cos(α + t) = cos(α) * cos(t) - sin(α) * sin(t)
sin(α + t) = sin(α) * cos(t) + cos(α) * sin(t)
Using these identities, we can rewrite the equations as:
x = -3 + 4 * cos(α + t)
y = 5 + 4 * sin(α + t)
Therefore, the rectangular form of the given parametric equations is:
(x + 3)^2 + (y - 5)^2 = 16
This equation represents the circle with a center (-3, 5) and a radius of 4.