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Chapter 6: Problem 51

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

Short Answer

Expert verified
The rectangular equation of the ellipse is \((x鈭抙)^2/a^2 + (y鈭択)^2/b^2 = 1\).

Step by step solution

01

Set up the equations

Given the parametric equations of an ellipse, \(x=h+a \cos \theta\) and \(y=k+b \sin \theta\). We first rearrange these equations to isolate \(\cos \theta\) and \(\sin \theta\). This gives us, \(\cos \theta = (x-h)/a\) and \(\sin \theta = (y-k)/b\).
02

Use the trigonometric identity

Now, we need to get rid of the parameters \(\cos \theta\) and \(\sin \theta\). We do this by applying the Pythagorean trigonometric identity \(\cos^2 \theta + \sin^2 \theta = 1\). Substituting from step 1, yields \(((x-h)/a)^2 + ((y-k)/b)^2 = 1\).
03

Write the final equation

Clean this equation up to write it in the standard form of an ellipse equation: \((x鈭抙)^2/a^2 + (y鈭択)^2/b^2 = 1\). This is the rectangular equation of the ellipse.

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