Problem 39

Question

Eliminate the parameter and obtain the standard form of the rectangular equation. Ellipse: $x=h+a \cos \theta, y=k+b \sin \theta$

Step-by-Step Solution

Verified

Answer

The standard form of the rectangular equation for the ellipse is $(x-h)^2/a^2 + (y-k)^2/b^2 = 1$

1Step 1: Express cos theta and sin theta

First, express $\cos \theta$ and $\sin \theta$ in terms of $x$ and $y$, respectively. To be specific, $\cos \theta= (x-h)/a$ and $\sin \theta= (y-k)/b$.

2Step 2: Use the trigonometric Pythagorean identity

Plug the expressions from Step 1 into the trigonometric Pythagorean identity $\cos^2 \theta + \sin^2 \theta = 1$. We get: $(x-h)^2/a^2 + (y-k)^2/b^2 = 1$

3Step 3: Obtain the standard form

The equation $(x-h)^2/a^2 + (y-k)^2/b^2 = 1$ is the standard form for the ellipse equation in rectangular coordinates.

Problem 39

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