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Chapter 10: Problem 141

Find the equation of a circle satisfying the given conditions. Center: (-8,-5)\(;\) radius: \(\sqrt{5}\)

Short Answer

Expert verified
(x + 8)^2 + (y + 5)^2 = 5

Step by step solution

01

Understand the standard form of a circle's equation

The standard equation of a circle with center \(h, k\) and radius \(r\) is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \]
02

Identify the values for h, k, and r

From the given conditions, the center of the circle \(h, k\) is (-8, -5) and the radius \(r\) is \sqrt{5}\.
03

Substitute the values into the standard equation

Substitute \[ h = -8 \], \[ k = -5 \], and \[ r = \sqrt{5} \] into the formula \[ (x - h)^2 + (y - k)^2 = r^2 \]. This gives: \[ (x - (-8))^2 + (y - (-5))^2 = ( \sqrt{5} )^2 \]
04

Simplify the equation

Simplify the equation obtained: \[ (x + 8)^2 + (y + 5)^2 = 5 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form of a Circle
The equation of a circle can be written in a very specific format known as the standard form. One can define a circle in a mathematical plane using just its center and its radius. The standard form of a circle's equation is given by:
\((x - h)^2 + (y - k)^2 = r^2 \).
This formula helps to graph and understand the circle's location and size easily. Here:
  • \

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