Problem 24 - 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. $$(-3,-2) \text { and }(1,-4)$$

Step-by-Step Solution

Answer

The circle's equation is \((x + 1)^2 + (y + 3)^2 = 5\).

Step 1: Find the Midpoint of the Diameter

To find the midpoint of the diameter, we use the midpoint formula which is \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Substitute \( x_1 = -3, y_1 = -2, x_2 = 1, \text{ and } y_2 = -4 \) into the formula: \[ M = \left( \frac{-3 + 1}{2}, \frac{-2 + (-4)}{2} \right) = \left( \frac{-2}{2}, \frac{-6}{2} \right) \] Thus, the midpoint (center of the circle) is \( (-1, -3) \).

Step 2: Calculate the Radius

The radius is the distance from the center to one of the endpoints of the diameter. We use the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Here, the center \((x_1, y_1) = (-1, -3)\) and an endpoint \((x_2, y_2) = (-3, -2)\). \[ \text{Radius} = \sqrt{(-3 + 1)^2 + (-2 + 3)^2} = \sqrt{(-2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \] So, the radius length is \( \sqrt{5} \).

Step 3: Form the Equation of the Circle

The equation of a circle in center-radius form is given by \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. We have \((h, k) = (-1, -3)\) and \(r = \sqrt{5}\). Therefore, the equation becomes: \[ (x + 1)^2 + (y + 3)^2 = (\sqrt{5})^2 \] Simplifying, we get: \[ (x + 1)^2 + (y + 3)^2 = 5 \] This is the equation of the circle.

Key Concepts

Midpoint Formula

The midpoint formula is an essential tool in geometry used to find the midpoint or the center point between two given points on a coordinate plane. It is especially useful when working with circles, as the midpoint of the endpoints of a diameter reveals the circle's center. The formula is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] To apply the midpoint formula, simply add the x-coordinates together and divide by 2. Do the same for the y-coordinates. This will give you the exact point that is centered between the two endpoints.

• For example, with endpoints at \((-3, -2)\) and \((1, -4)\), we calculate:- \[ \frac{-3 + 1}{2} = \frac{-2}{2} = -1 \]- \[ \frac{-2 + (-4)}{2} = \frac{-6}{2} = -3 \]
• Thus, the midpoint, which becomes the circle's center, is \((-1, -3)\).

Radius Calculation

Once you have the circle's center using the midpoint formula, the next step is calculating the radius. The radius is the distance from the center of the circle to any point on its circumference. If you know the center and any endpoint of the diameter, you can easily calculate the radius using the distance formula. The formula is:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Where:

• \((x_1, y_1)\) is the center
• \((x_2, y_2)\) is one endpoint of the diameter.

In our example, the center is \((-1, -3)\), and you can choose endpoint \((-3, -2)\):

• \[ \text{Radius} = \sqrt{(-3 + 1)^2 + (-2 + 3)^2} = \sqrt{(-2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
• Thus, the radius of the circle is \( \sqrt{5} \).

Distance Formula

The distance formula finds the distance between two points in a coordinate plane and is key for calculating the radius of a circle when the center and a point on the circle are known. The formula is similar to the Pythagorean Theorem and is expressed as: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula helps you measure exact lengths between two points:

• Subtract the x-coordinates and square the result.
• Subtract the y-coordinates and square the result.
• Add those two squared numbers together, then take the square root of that sum.

In a circle scenario, use this formula to find how far the center is from a given endpoint, helping confirm the circle's radius.

Center-Radius Form

The center-radius form of a circle's equation is a vital equation for describing a circle in a two-dimensional plane. With two essential parts—the circle's center and its radius—it is represented as: \[(x-h)^2 + (y-k)^2 = r^2\] Where:

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

In practice, start by plotting \((h, k)\) and use \(r\) to determine the size of the circle. In our example with center \((-1, -3)\) and radius \(\sqrt{5}\), the equation is:

• \[ (x + 1)^2 + (y + 3)^2 = 5 \]

This simple equation lets anyone instantly know both the location and size of the circle, making it easier to graph and analyze.