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Chapter 1: Problem 79

How is the standard form of a circle's equation obtained from its general form?

Short Answer

Expert verified
The standard form of a circle's equation, \( (x - h)^2 + (y - k)^2 = r^2 \) is obtained from its general form, \(Ax^2 + By^2 + Cx + Dy + E = 0\), by completing the square. Here, h = -C/2A, k = -D/2B, and \( r = \sqrt{(C/2A)^2+(D/2B)^2 - E} \).

Step by step solution

01

General Form of a Circle Equation

The general form of a circle's equation is \(Ax^2 + By^2 + Cx + Dy + E = 0\), where A equals B, and A and B are both not equal to zero.
02

Completing the Square

We need to complete the square in this scenario. Group x's and y's together, the general form becomes \((A)x^2 + Cx + (B)y^2 + Dy + E = 0\). After that, rewrite it as \((A)(x^2 + (C/A)x) + (B)(y^2 + (D/B)y) = -E)\.
03

Transform to Standard Form

You can complete the square in the x's and y's terms by adding \( (C/2A)^2 \) for x's and \( (D/2B)^2 \) for y's on both sides of the equation respectively. Thus, the equation transforms into its standard form \((x - h)^2 + (y - k)^2 = r^2\), where h = -C/2A, k = -D/2B, and \( r = \sqrt{(C/2A)^2+(D/2B)^2 - E} \).

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Most popular questions from this chapter

?. The regular price of a pair of jeans is \(x\) dollars. Let \(f(x)=x-5\) and \(g(x)=0.6 x\) a. Describe what functions \(f\) and \(g\) model in terms of the price of the jeans. b. Find \((f \circ g)(x)\) and describe what this models in terms of the price of the jeans. c. Repeat part (b) for \((g \circ f)(x)\) d. Which composite function models the greater discount on the jeans, \(f \circ g\) or \(g \circ f ?\) Explain.

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