Answer
Cartesian Equation: $4x^2 + \frac 1 4 y^2 = 1$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/98467b05-68fe-49ad-a3ed-237b9da044d1/result_image/1550773371.png?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20250215%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20250215T144413Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=bfcf71d599675870d16e88368f32c016e9487921e0fcbbb4a7568315588c86fa)
Work Step by Step
(a)
$x = \frac 1 2 cos \theta \longrightarrow x^2 = (\frac 1 2 cos\theta)^2 = \frac 1 4 cos ^2 \theta \longrightarrow 4x^2 =cos^2 \theta$
$y = 2 sin \theta \longrightarrow y^2 = 4 sin^2 \theta \longrightarrow \frac 1 4 y^2 = sin^2 \theta$
Adding the equations:
$4x^2 + \frac 1 4 y^2 = cos^2 \theta + sin^2 \theta$
$4x^2 + \frac 1 4 y^2 = 1$
(b)
1. Plot points determined by values for $\theta$ between $0$ and $\pi$.
2. Join them to produce a curve.
3. Draw the arrows indicating which direction the curve goes from $\theta = 0$ to $\theta = \pi$