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Chapter 1: Problem 42

Write the standard equation for each circle. Center at \((0,0)\) with radius 5

Short Answer

Expert verified
The standard equation is \( x^2 + y^2 = 25 \).

Step by step solution

01

Identify the general form of the circle equation

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

Substitute the center coordinates

Plug in \( h = 0 \) and \( k = 0 \) into the general form of the circle equation: \[ (x-0)^2 + (y-0)^2 = r^2 \].
03

Simplify the equation with the given radius

Simplify the equation \[ (x-0)^2 + (y-0)^2 = r^2 \] by replacing \( r \) with 5: \[ x^2 + y^2 = 5^2 \].
04

Calculate the square of the radius

Compute \( 5^2 \) to get \( 25 \). Therefore, the equation becomes \( x^2 + y^2 = 25 \).

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Key Concepts

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

circle center coordinates
The center of a circle is a crucial point from which all parts of the circle are equidistant. It is denoted by coordinates \((h, k)\) in the circle's equation.
A circle's center can be easily identified by looking at the equation and finding values for \(h\) and \(k\).
For example, in the given exercise, the center coordinates are \((0,0)\).
radius of a circle
The radius of a circle is the distance from its center to any point on the circle. It is commonly represented by the letter \(r\).
The radius is an essential part of the circle's equation because it defines the size of the circle.
In the problem above, the radius is given as 5. When using the radius in the equation, you need to square the radius value to fit the standard form. Hence, \(r^2 = 25\).
standard form of a circle equation
The standard form of a circle equation combines the center coordinates and the radius into a simple mathematical formula.
The formula is: \[ (x-h)^2 + (y-k)^2 = r^2 \]
where \( (h,k)\) are the center coordinates and \(r\) is the radius.
In the given example, substituting \((h, k) = (0, 0)\) and \(r = 5\) into the formula gives you:
\[ (x-0)^2 + (y-0)^2 = 5^2 \].
Simplifying this, you get \[ x^2 + y^2 = 25 \], which is the standard form of the circle equation for the given problem.

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