Question

Consider the following. x = 7 cos θ, y = 8 sin θ, −π/2 ≤ θ ≤ π/2

(a) Eliminate the parameter to find a Cartesian equation of the curve.

Answer 1

To eliminate the parameter θ and find the Cartesian equation of the given curve, rewrite the equations x = 7cos(θ) and y = 8sin(θ) in terms of x and y. Use the trigonometric identity cos²(θ) + sin²(θ) = 1 to eliminate θ. The resulting equation x²/49 + y²/64 = 1 represents an ellipse with center at the origin and semi-major and semi-minor axes of lengths 8 and 7, respectively.

Step-by-step explanation:

To eliminate the parameter and find a Cartesian equation of the given curve, we can rewrite the equations in terms of x and y. Since x = 7cos(θ) and y = 8sin(θ), we can use the trigonometric identity cos²(θ) + sin²(θ) = 1 to eliminate θ. Squaring both x and y equations and dividing them by 49 and 64, respectively, we get:

x²/49 + y²/64 = 1

This equation represents an ellipse with center at the origin, semi-major axis of length 8, and semi-minor axis of length 7.