Problem 25 - A Graphical Approach to College Algebra

Question

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius, which is the distance from the center to either endpoint of the diameter. Finally use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. $$(-5,0) \text { and }(5,0)$$

Step-by-Step Solution

Answer

The equation of the circle is \(x^2 + y^2 = 25\).

Step 1: Find the Midpoint of the Diameter

To find the midpoint of the diameter with endpoints \((-5,0)\) and \((5,0)\), use the midpoint formula: \((x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Insert the given points: \((x_m, y_m) = \left(\frac{-5 + 5}{2}, \frac{0 + 0}{2}\right) = (0, 0)\). Thus, the midpoint (center of the circle) is \((0,0)\).

Step 2: Calculate the Radius

The radius is the distance from the center \((0,0)\) to one of the endpoints, which is either \((-5,0)\) or \((5,0)\). Use the distance formula: \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). For \((0,0)\) to \((5,0)\): \(r = \sqrt{(5 - 0)^2 + (0 - 0)^2} = \sqrt{25} = 5\). Thus, the radius is 5.

Step 3: Write the Equation in Center-Radius Form

The equation of a circle in center-radius form is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Plug in the center \((0,0)\) and the radius \(5\): \((x - 0)^2 + (y - 0)^2 = 5^2\). Simplify to get: \(x^2 + y^2 = 25\).

Key Concepts

Midpoint Formula

The midpoint formula is a crucial tool in geometry for finding the exact middle point of a line segment. This point is often referred to as the midpoint. It is particularly useful when determining the center of a circle given the endpoints of its diameter.

To calculate the midpoint, you use the formula:

• Midpoint \( (x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \)

This formula averages the x-coordinates and the y-coordinates of the endpoints. In our example, the endpoints of the diameter are \( (-5,0) \) and \( (5,0) \). Applying the formula:

• Midpoint \( (x_m, y_m) = \left(\frac{-5 + 5}{2}, \frac{0 + 0}{2}\right) = (0, 0) \)

Thus, the center of our circle is at \( (0, 0) \). The midpoint gives us the starting point to calculate the radius as well as craft the circle's equation.

Radius Calculation

The radius of a circle is a line segment from the center of the circle to any point on its circumference. It directly influences the size of the circle. Knowing the radius is essential for writing the equation of a circle, especially in center-radius form.

To find the radius, use the distance formula between the circle's center and one of its points on the diameter:

• Radius \( r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

For example, if we have our center \( (0,0) \) and one endpoint of the diameter \( (5,0) \), the calculation is:

• Radius \( r = \sqrt{(5-0)^2 + (0-0)^2} = \sqrt{25} = 5 \)

The computed radius is 5. This corresponds to half the length of the diameter, perfecting the circle’s shape around its center.

Center-Radius Form

In order to write the equation of a circle, we employ the center-radius form. This form elegantly describes the circle as it incorporates both the center and the radius into a neat equation. The general formula is:

• Center-Radius Form: \( (x - h)^2 + (y - k)^2 = r^2 \)

Here, \( h \) and \( k \) represent the x and y coordinates of the center, while \( r \) represents the radius. Substituting the center \( (0,0) \) and radius \( 5 \) into this form gives:

• Circle Equation: \( (x - 0)^2 + (y - 0)^2 = 5^2 \)

This simplifies to: \( x^2 + y^2 = 25 \). This equation represents all points equidistant from the center, effectively defining the circle. Understanding this form is foundational when dealing with circle equations in geometry.