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Chapter 10: Problem 28

Sketch the circle. Identify its center and radius. $$y^{2}=81-x^{2}$$

Short Answer

Expert verified
The circle is centered at the origin (0,0) and has a radius of 9 units.

Step by step solution

01

Rearrange the given equation

Start by rearranging the given equation \(y^2 = 81 - x^2\) to bring it in the form of a standard circle equation. This can be done by adding \(x^2\) to both sides to get \(x^2 + y^2 = 81\).
02

Identify the center

With the equation now in the form \(x^2 + y^2 = 81\), we can see that the circle is centered at the origin (0,0). This is because there's no constant added or subtracted to \(x\) or \(y\) in the equation.
03

Identify the radius

The given circle鈥檚 equation \(x^2 + y^2 = 81\) is in the standard form where 81 represents \(r^2\). To determine the radius, simply take the square root of 81. This gives a radius of 9.
04

Sketch the circle

Now you can sketch the circle. Draw a point at the origin (0,0), this is the center of the circle. Then, using a compass or a circle drawing tool, draw a circle with a radius of 9 units. You can measure 9 units in any direction from the center to draw this circle. As a result, an accurate sketch of the circle should be a perfect circle centered at the origin with a radius of 9 units.

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Key Concepts

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

Center of a Circle
The center of a circle is a crucial point that defines its position in a coordinate plane. In a circle's equation, the center is represented by the point \(h, k\). For example, in the standard form equation \( (x - h)^2 + (y - k)^2 = r^2 \), \(h\) and \(k\) are the coordinates of the circle's center. If these values are zero, as seen in \(x^2 + y^2 = 81\), the circle is centered at the origin, (0,0).

  • Centering a circle at (0,0) simplifies its equation to \(x^2 + y^2 = r^2\)
  • The center acts as a fixed point around which the circle is perfectly symmetrical
Understanding the center helps in visualizing and sketching the circle accurately.
Radius of a Circle
The radius is the distance from the center of the circle to any point on its edge. It is a constant length that defines the size of the circle. In the equation \( (x - h)^2 + (y - k)^2 = r^2 \), \(r^2\) is the radius squared.

For the equation \(x^2 + y^2 = 81\), this means \(r^2 = 81\). Taking the square root of both sides, \(r = \sqrt{81} = 9\). Therefore, the radius is 9.

  • The radius is vital for determining the circle's size and extent
  • Every point on the circle is equidistant from the center, making the radius a uniform measure
Understanding the radius is essential for sketching and spatial awareness of the circle's dimensions.
Standard Form of a Circle Equation
The standard form of a circle equation is a useful representation that makes it easy to identify a circle's center and radius. This form is written as \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \(r\) is the radius.

The exercise transformed \(y^2 = 81 - x^2\) into \(x^2 + y^2 = 81\), aligning it with the standard form where \((h,k) = (0,0)\) and \(r^2 = 81\).

  • Standard form allows for quick identification of the circle's main features
  • It is especially handy for graphing and analyzing circles on the coordinate plane
Mastering this form simplifies many geometrical problems involving circles.

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Most popular questions from this chapter

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