Problem 21 - A Graphical Approach to Precalculus with Limits

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. $$(-1,3) \text { and }(5,-9)$$

Step-by-Step Solution

Answer

The equation is \((x - 2)^2 + (y + 3)^2 = 45\).

Step 1: Find the Midpoint

To find the midpoint of the diameter, use the midpoint formula \(M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). Given the points \((-1, 3)\) and \((5, -9)\), calculate:\[x = \frac{-1 + 5}{2} = 2\]\[y = \frac{3 + (-9)}{2} = -3\]Thus, the midpoint (and center of the circle) is \((2, -3)\).

Step 2: Calculate the Radius

The radius is the distance from the center \((2, -3)\) to one of the endpoints, say \((-1, 3)\). Use the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substitute the values:\[d = \sqrt{(2 - (-1))^2 + (-3 - 3)^2} = \sqrt{3^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45}\]Thus, the radius is \(\sqrt{45} = 3\sqrt{5}\).

Step 3: Write the Equation of the Circle

Use the center-radius form of a circle equation: \[(x - h)^2 + (y - k)^2 = r^2\]Where \((h, k)\) is the center and \(r\) the radius. Substitute \((h, k) = (2, -3)\) and \(r = 3\sqrt{5}\):\[(x - 2)^2 + (y + 3)^2 = (3\sqrt{5})^2\]Simplify:\[(x - 2)^2 + (y + 3)^2 = 45\]

Key Concepts

Midpoint Formula

In geometry, the midpoint formula is a powerful tool for finding the exact center point between two defined endpoints on a coordinate plane. This can be particularly useful when working with geometric shapes like circles. To find the midpoint between two points, use this formula:

• Midpoint, \( M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

Given the points \((-1, 3)\) and \((5, -9)\), we find the midpoint by calculating:

• For \(x\): \( \frac{-1 + 5}{2} = 2 \)
• For \(y\): \( \frac{3 + (-9)}{2} = -3 \)

Hence, the midpoint is \((2, -3)\), which is also the center of the circle this diameter belongs to.
Remember, the midpoint divides the diameter into two equal segments and always lies directly between the two endpoints.

Distance Formula

The distance formula helps us determine the distance between two points on a coordinate plane, which is crucial in calculating the radius of our circle. The radius is defined as the length from the center of the circle to any point on its circumference. Using the endpoint \((-1, 3)\) and the center \((2, -3)\), we find the distance (radius) as follows:

• Distance Formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

Substitute the coordinates:

• \( d = \sqrt{(2 - (-1))^2 + (-3 - 3)^2} = \sqrt{3^2 + (-6)^2} \)
• This simplifies to \( \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} \)

Therefore, the radius of the circle is \(3\sqrt{5}\). This formula is derived from the Pythagorean theorem and is used extensively in geometry to compute lengths and distances.

Center-Radius Form

The center-radius form of a circle's equation is a convenient way to express the properties of a circle with its center and radius. The standard formula is:

• \( (x - h)^2 + (y - k)^2 = r^2 \)

Here:

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

Using our previous results, where the center \((h, k) = (2, -3)\) and radius \(r = 3\sqrt{5}\), we formulate the equation:

• \((x - 2)^2 + (y + 3)^2 = (3\sqrt{5})^2\)
• Which simplifies to \((x - 2)^2 + (y + 3)^2 = 45\)

This form not only defines the geometric nature of the circle but also makes it easy to graph on a coordinate plane, showing at a glance the circle's central location and the spread of its radius.