Consider the parametric equations

x = a cos t + b sin t, y = c cos t + d sin t,

where a, b, c, and d are real numbers.

a. Show that (apart from a set of special cases) the equations describe an ellipse of the form Ax² + Bxy + Cy² = K, where A, B, C, and K are constants.

b. Show that (apart from a set of special cases), the equations describe an ellipse with its axes aligned with the x- and y-axes provided ab + cd = 0.

c. Show that the equations describe a circle provided ab + cd = 0 and c² + d² = a² + b² ≠ 0.