/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 137 Find the equation of a circle sa... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 137

Find the equation of a circle satisfying the given conditions. Center: (0,0)\(;\) radius: 12

Short Answer

Expert verified
The equation of the circle is \( x^2 + y^2 = 144 \).

Step by step solution

01

Identify the formula for a circle

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

Substitute the center coordinates

Given the center \(0,0\), it means \(h = 0\) and \(k = 0\). Substitute these into the general circle equation: \[ (x - 0)^2 + (y - 0)^2 = r^2 \]
03

Simplify the equation

Simplify the equation derived from substituting the center coordinates: \[ x^2 + y^2 = r^2 \]
04

Substitute the radius

Given the radius is 12, which means \(r = 12\). Substitute this value into the simplified equation: \[ x^2 + y^2 = 12^2 \]
05

Calculate the radius squared

Calculate \(12^2\): \[ 12^2 = 144 \]
06

Write the final equation

Substitute \(144\) back into the previous equation to get the final circle equation: \[ x^2 + y^2 = 144 \]

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Key Concepts

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

circle equation
The equation of a circle is crucial to understanding its geometry. It allows us to determine the points that lie on the circumference of the circle. The standard form of a circle鈥檚 equation is given by the formula: \[ (x - h)^2 + (y - k)^2 = r^2 \]In this formula:
  • **(h, k)** are the coordinates of the circle's center
  • **r** is the radius of the circle
In our exercise, the center of the circle is at (0,0), simplifying our equation since both h and k are zero.
radius
The radius of a circle is the distance from its center to any point on its circumference. It is a crucial component of the circle equation. In the circle equation \((x - h)^2 + (y - k)^2 = r^2\), **r** represents the radius.
For our example, the radius is given as 12. This means we substitute **r** with 12, and thus, \(r^2\) equals \(12^2\), which is 144. So the equation becomes:\(x^2 + y^2 = 144\)
center coordinates
The center of a circle is defined by the coordinates \((h, k).\) These coordinates are vital as they determine the positioning of the circle on the coordinate plane. In the general formula \((x - h)^2 + (y - k)^2 = r^2\),

Substituting the center coordinates

Given that our center is at \((0, 0),\) substituting **h** and **k** with 0, we get \[(x - 0)^2 + (y - 0)^2 = r^2.\] Simplifying this:\[x^2 + y^2 = r^2.\]This equation is now ready for the radius to be added in, which completes the formula and positions your circle on the plane accurately.

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