Question

x=2 cos(t), y=3 sin(t), 0 ≤ t ≤ 2π A pair of parametric equations is given. Find a rectangular-coordinate equation for the curve by eliminating the parameter.

Answer 1

To eliminate the parameter t from the given parametric equations and find the rectangular-coordinate equation, we use trigonometric identities to derive that the curve is an ellipse given by the equation x²/4 + y²/9 = 1.

Step-by-step explanation:

To find a rectangular-coordinate equation for the curve given by the parametric equations x=2 cos(t), y=3 sin(t), where 0 ≤ t ≤ 2π, we can use the Pythagorean trigonometric identity sin2(t) + cos2(t) = 1.

Dividing x and y equations by their respective coefficients, we have cos(t) = x/2 and sin(t) = y/3. Squaring both sides of these equations and then summing, we get (x/2)2 + (y/3)2 = cos2(t) + sin2(t) = 1. Simplifying, the rectangular-coordinate equation is x2/4 + y2/9 = 1, which represents an ellipse.