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Chapter 10: Problem 38

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

Short Answer

Expert verified
The standard form of the rectangular equation of the circle is \((x-h)^2 + (y-k)^2 = r^2\).

Step by step solution

01

Express \(x-h\) and \(y-k\) in terms of \(r\)

Start by expressing \(x-h\) and \(y-k\) as: \(x-h = r \cos \theta\) and \(y-k = r \sin \theta\), isolating the cosine and sine terms.
02

Square each equation

Square both equations to obtain: \((x-h)^2 = (r \cos \theta)^2\) and \((y-k)^2 = (r \sin \theta)^2\). This gives: \((x-h)^2 = r^2 \cos^2 \theta\) and \((y-k)^2 = r^2 \sin^2 \theta\)
03

Add the squared equations

Add the two equations to express the left-hand side as a sum of squares: \((x-h)^2 + (y-k)^2 = r^2 \cos^2 \theta + r^2 \sin^2 \theta\)
04

Apply the Pythagorean identity

Use the Pythagorean identity \(\cos^2{\theta} + \sin^2{\theta}=1\) to replace the right-hand side of the equation, resulting in: \((x-h)^2 + (y-k)^2 = r^2( \cos^2 \theta + \sin^2 \theta) = r^2\)

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Most popular questions from this chapter

Convert the polar equation to rectangular form. $$r=\frac{6}{2-3 \sin \theta}$$

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