Problem 18 - Algebra for College Students

Question

For Problems \(15-32\), find the center and the length of a radius of each of the circles. $$ (x-7)^{2}+(y+2)^{2}=24 \quad(7,-2) ; r=2 \sqrt{6} $$

Step-by-Step Solution

Answer

The center is (7, -2) and the radius is \(2\sqrt{6}\).

Step 1: Identify the General Equation of a Circle

The general equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Compare Given Equation with General Form

The given equation is \((x-7)^2 + (y+2)^2 = 24\). Compare this with the general form \((x-h)^2 + (y-k)^2 = r^2\). From this comparison, we identify \(h = 7\), \(k = -2\), and \(r^2 = 24\).

Step 3: Identify the Center of the Circle

Using the values of \(h\) and \(k\) from the comparison, the center of the circle is \((h, k) = (7, -2)\).

Step 4: Calculate the Radius

To find the radius \(r\), we need to take the square root of \(r^2\). So, \(r = \sqrt{24}\). Simplify this to \(r = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}\).

Key Concepts

General Equation of a Circle

Understanding the equation of a circle is key in many geometry problems. The general form of a circle's equation is \[(x-h)^2 + (y-k)^2 = r^2\]where:

• \((h, k)\) is the center of the circle.
• \(r\) is the radius of the circle.

This equation represents a circle in a coordinate plane, showing how every point \((x, y)\) that satisfies it is exactly \(r\) units away from the center point \((h, k)\). By comparing a circle's given equation to the general form, you can extract important geometric information like the circle's center and radius.

Circle Center

Finding the circle's center from its equation involves identifying the values \(h\) and \(k\) in the general circle equation. In an equation like:\[(x-7)^2 + (y+2)^2 = 24\], you can see:

• The expression \((x-7)^2\) means \(h = 7\).
• The expression \((y+2)^2\) implies \(k = -2\), because it can be rewritten as \((y-(-2))^2\).

Thus, the center of this particular circle is at the coordinates \((7, -2)\). This center is a fixed point in the plane from which all points on the circle are equidistant.

Circle Radius

Determining the radius is a straightforward process once you have the equation. In the given example \[(x-7)^2 + (y+2)^2 = 24\],

• The right side of the equation provides \(r^2 = 24\).
• To find the radius \(r\), take the square root of \(24\): \(r = \sqrt{24}\).

Simplifying further, notice that \(24 = 4 \times 6\), so

• \(r = \sqrt{4 \times 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}\).

Thus, the circle's radius is \(2\sqrt{6}\). This radius indicates the constant distance from the center \((7, -2)\) to any point on the circle.