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Chapter 11: Problem 8

Find the equation of a circle satisfying the given conditions. Center: \((5,-2) ;\) radius: 4

Short Answer

Expert verified
The equation of the circle is \( (x-5)^2 + (y+2)^2 = 16 \).

Step by step solution

01

Identify the General Form of Circle Equation

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

Substitute the Center Coordinates

Substitute \(h=5\) and \(k=-2\) into the general form: \[ (x-5)^2 + (y-(-2))^2 = r^2 \]
03

Simplify Inside the Parentheses

Simplify the equation inside the parentheses: \[ (x-5)^2 + (y+2)^2 = r^2 \]
04

Substitute the Radius

Substitute \(r=4\) into the equation: \[ (x-5)^2 + (y+2)^2 = 4^2 \]
05

Simplify the Radius Term

Calculate \(4^2\): \[ (x-5)^2 + (y+2)^2 = 16 \]

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Key Concepts

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

center of circle
To find the equation of a circle, one of the key components we need to know is the center of the circle. The center is a point \(h,k\) that the circle is equidistant from in all directions. In this problem, the center given is \(5, -2\). This means the circle is centered at the point where the x-coordinate is 5 and the y-coordinate is -2. This center point is vital because it helps us shift the equation of the circle accordingly. Plugging the coordinates of the center into the general form of the circle's equation adjusts the equation to correctly represent this specific circle.
radius
The radius of the circle is the distance from the center of the circle to any point on the circle. In this problem, the radius is given as 4 units. The radius is important because it determines the size of the circle. In the general form of the circle equation, the radius \(r\) is squared and placed on the right side of the equation: \[ (x - h)^2 + (y - k)^2 = r^2 \]. By substituting \(r = 4\), we get \[ (x - 5)^2 + (y + 2)^2 = 4^2 \.\] Therefore, in the final equation, we use \(16\) as the value for \(r^2\).
general form of circle equation
The general form of a circle's equation is a significant formula in coordinate geometry. It is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \]. Here, \(h\) and \(k\) are the coordinates of the circle's center, and \(r\) is the radius. This form is constructed to ensure that any point \((x, y)\) on the circle is exactly \(r\) units away from the center \((h, k)\). By substituting the specific values for \(h, k\), and \(r\) into this formula, we obtain a precise equation that represents our circle.

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