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Chapter 13: Problem 29

Find an equation of the circle satisfying the given conditions. Center \((0,0),\) radius 6

Short Answer

Expert verified
x^2 + y^2 = 36.

Step by step solution

01

Identify circle equation

..
02

Substitute given center

Given the center (0,0) , substitute \(h=0\) and \(k=0\) into the standard equation of the circle: ...
03

Substitute given radius

...

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Key Concepts

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

Standard equation of a circle
The equation of a circle is a fundamental concept in geometry. It helps us understand the shape and size of a circle on a coordinate plane. The standard form of the equation of a circle is:
\( (x - h)^2 + (y - k)^2 = r^2 \)
Here, \( (h, k) \) represents the center of the circle, and \( r \) is the radius.
This formula comes from the Pythagorean theorem. It means that any point \( (x, y) \) on the circle is at a distance \( r \) from its center. It's like drawing a circle using a radius as the distance from the center to the edge.
Center of a circle
The center of a circle is a crucial part of its equation. In the standard circle equation, \( (h, k) \) are the coordinates. Here鈥檚 how you use those values:
  • If the center is at the origin, then \( h = 0 \) and \( k = 0 \).
  • For example, the center of our circle is \( (0,0) \).
  • Substituting these into the equation, we get \( (x - 0)^2 + (y - 0)^2 = r^2 \).
This simplifies to:
\( x^2 + y^2 = r^2 \).
Therefore, when the center is at the origin, the equation is simpler. It's just the sum of the squares of \( x \) and \( y \) equal to the square of the radius.
Radius of a circle
The radius of a circle is the distance from the center to any point on its edge. You denote it by \( r \) in the circle equation. Here鈥檚 how you involve the radius in the calculation:
  • In our problem, the radius is 6.
  • So, we substitute \( r = 6 \) into the simplified equation we got earlier.
  • This gives us: \( x^2 + y^2 = 6^2 \).
Simplifying, we get:
\( x^2 + y^2 = 36 \).
This is the final equation of the circle with center \( (0,0) \) and radius 6. By knowing how to substitute the values of the center and radius into the equation, you can describe any circle.

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