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Chapter 7: Problem 138

Find the equation of a circle satisfying the given conditions. Center: (0,0)\(;\) radius: 9

Short Answer

Expert verified
The equation of the circle is \[ x^2 + y^2 = 81 \].

Step by step solution

01

Identify the general form of the circle's equation

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

Substitute the center coordinates

Given the center of the circle is \(0, 0\), substitute \(h = 0\) and \(k = 0\) into the general form equation: \[ (x - 0)^2 + (y - 0)^2 = r^2 \] Simplifies to: \[ x^2 + y^2 = r^2 \]
03

Substitute the radius

Given the radius of the circle is \(9\), substitute \(r = 9\) into the simplified equation: \[ x^2 + y^2 = 9^2 \] Which simplifies to: \[ x^2 + y^2 = 81 \]
04

Write the final equation

The equation of the circle is: \[ x^2 + y^2 = 81 \]

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Key Concepts

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

Center-Radius Form
To find the equation of a circle, it's essential to first understand the center-radius form of the equation. The center-radius form of a circle's equation is: \[ (x - h)^2 + (y - k)^2 = r^2 \] where
  • \( h, k \) represent the coordinates of the center of the circle.
  • \( r \) is the radius of the circle.
By identifying the center and the radius, you can easily plug these values into the center-radius form and find the equation of the circle. This form is very useful as it directly shows the circle's center and radius, making it easy to understand and visualize the circle.
Substitute Coordinates
Having identified the center and radius, the next step is to substitute the center coordinates into the general form of the circle's equation: \[ (x - h)^2 + (y - k)^2 = r^2 \] Given the center is at \( (0,0) \), we substitute \( h = 0 \) and \( k = 0 \): \[ (x - 0)^2 + (y - 0)^2 = r^2 \] After substitution, the equation simplifies as follows: \[ x^2 + y^2 = r^2 \] So after substituting the center coordinates, we see that the circle's equation simplifies to a form without any subtracted terms, making it easier to work with.
Simplified Equation
Next, we proceed to simplify the equation further by substituting the radius value. For our circle, the given radius is \( 9 \).By substituting \( r = 9 \) into the simplified equation \( x^2 + y^2 = r^2 \): \[ x^2 + y^2 = 9^2 \] Calculating \( 9^2 \) gives us: \[ x^2 + y^2 = 81 \] This is the simplified equation of the circle that has all terms clearly defined and no further calculations needed. It鈥檚 always advisable to square the radius first to avoid any confusion.
General Form of Circle
Finally, let's look at the general form of a circle's equation. The general form is typically written as: \[ x^2 + y^2 + Dx + Ey + F = 0 \] This is useful for rewriting the center-radius form in a way that can be easier to handle in certain algebraic manipulations. When we have \[ x^2 + y^2 = 81 \], we can state that:
  • \( D = 0 \)
  • \( E = 0 \)
  • \( F = -81 \)
So, \[ x^2 + y^2 - 81 = 0 \] is the general form of our circle's equation. This form offers an alternative representation that can help in comparing different circles or solving systems of equations involving circles.

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