Cartesian Equation and Parametric

Question

We were given this and the cartesian equation of ( (x^2/ a^2) + (y^2/b^2)) = 1. How would I solve for a, b, the initial point, and the terminal point?

Answer 1

To find the initial and terminal points, use the endpoints of the domain interval given for t:

x(-π/2) = 0 , y(-π/2) = -8 so initial pt: (0,-8) ; x(π/2) = 0 , y(π/2) = 8 so terminal pt: (0,8)

To convert to cartesian form, divide both sides of the equation given for x by 13 and divide both sides of the equation given for y by 8. Square both and add them, using the fundamental trig ID: sin²θ + cos²θ = 1.

x/13² + y/8² = 1 ; so a = 13 and b = 8.

From the initial and terminal pts above, we conclude this set of parametric equations will graph (from bottom to top) the right-hand side (only) of a horizontal ellipse with major axis of length 26 and minor axis of length 16.