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Chapter 6: Problem 50

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 for the circle is \( (x-h)^2 + (y-k)^2 = r^2 \).

Step by step solution

01

Square both equations and add them together

Squaring both the provided equations yields \( x^2 = h^2 + 2hr \cos \theta + r^2 \cos^2 \theta \) and \( y^2 = k^2 + 2kr \sin \theta + r^2 \sin^2 \theta \). Now add both these equations together.
02

Apply trigonometric identity

The trigonometric identity, \( \cos^2 \theta + \sin^2 \theta = 1 \), is then used in the equation obtained in Step 1: \( x^2 + y^2 = h^2 + k^2 + 2hr \cos \theta + 2kr \sin \theta + r^2 \).
03

Simplify equation

Simplify the equation by cancelling out the terms \( 2hr \cos \theta \) and \( 2kr \sin \theta \) because they will equal 0 due to adding from step 1. This leaves the equation as \( x^2 + y^2 = h^2 + k^2 + r^2 \).
04

Rearrange into standard form

Rearrange the equation into a standard circular equation form, obtaining \( (x-h)^2 + (y-k)^2 = r^2 \).

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