Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(x^2 + y^2 = 81\).

Step 2: 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.

Step 3: Identify the radius

The given circle’s 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.

Step 4: 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.

Key Concepts

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.