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Chapter 10: Problem 39

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

Step by step solution

01

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 \(\sin \theta= (y-k)/b\).
02

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: \((x-h)^2/a^2 + (y-k)^2/b^2 = 1\)
03

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.

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Most popular questions from this chapter

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{8}{4+3 \sin \theta}$$

Convert the polar equation to rectangular form. $$r=4 \sin \theta$$

Convert the polar equation to rectangular form. $$r=2 \cos \theta$$

Convert the rectangular equation to polar form. Assume \(a<0\) $$3 x+5 y-2=0$$

Show that the polar equation of the ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad \text { is } \quad r^{2}=\frac{b^{2}}{1-e^{2} \cos ^{2} \theta}$$
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