Find a parameterization for a circle of radius 2 with center (-2,6,-3) in a plane parallel to the xz plane. Write your parameterization so the x component includes a positive cosine.

Answer 1

All the points in a plane parallel to the xz plane share the same y-coordinate.

Since the circle's center has y-coordinate 6, all the points of that circle have

y = 6

Consider any point P on the circle.

Let θ be the angle between the positive x-axis and the line connecting P with the center.

Then the x-coordinate of the point P is -2 + 2cos(θ), and

the z-coordinate is -3 + 2sin(θ).

So the parametrization of the circle is

x = -2 + 2cos(θ)

y = 6

z = -3 + 2sin(θ)

for 0 ≤ θ < 2π