a)
x = 2√2 cos θ
y = 2 sin θ
x = 2√2 cos θ
Square both sides of this equation
x2 = 8 cos2 θ
Divide both sides by 8.
x2 / 8 = cos2 θ
y = 2 sin θ
Square both sides of this equation.
y2 = 4 sin2 θ
By the Pythagorean identity, we can substitute (1 - cos2θ) in for sin2θ
y2 = 4(1 - cos2θ)
Substitute x2 / 8 in for cos2θ
y2 = 4(1 - x2 / 8)
Divide both sides of the equation by 4 .
y2 / 4 = 1 - x2 / 8
Add x2 / 8 to both sides of the equation.
x2 / 8 + y2 / 4 = 1
Here is what the ellipse looks like:
https://www.desmos.com/calculator/o8fvqgkljx
the distance from the center of the ellipse to a focus = c
c2 = a2 - b2 = 8 - 4 = 4
c = √4 = 2
the foci are located at (0 + 2, 0) and (0 - 2, 0)
the foci are located at (2, 0) and (-2, 0)