Problem 22 - 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. $$(-4,5) \text { and }(6,-9)$$

Step-by-Step Solution

Answer

The equation of the circle is \((x-1)^2 + (y+2)^2 = 74\).

Step 1: Finding the Center of the Circle

The center of the circle is the midpoint of the diameter. To find the midpoint between the two given endpoints, \((-4,5)\) and \((6,-9)\), use the midpoint formula: \(M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\). Substituting the given values, \(M = \left(\frac{-4+6}{2}, \frac{5-9}{2}\right) = (1, -2)\). So, \((1, -2)\) is the center of the circle.

Step 2: Calculating the Radius

To find the radius, calculate the distance from the center \((1, -2)\) to one of the endpoints of the diameter, say \((-4, 5)\), using the distance formula: \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Therefore, \(r = \sqrt{((-4 - 1)^2 + (5 + 2)^2)} \). Solving, \(r = \sqrt{(25 + 49)} = \sqrt{74}\). So, the radius is \(\sqrt{74}\).

Step 3: Writing the Equation in Center-Radius Form

The equation for 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. From previous steps, \((h, k) = (1, -2)\) and \(r = \sqrt{74}\). Therefore, the equation of the circle is \((x-1)^2 + (y+2)^2 = 74\).

Key Concepts

Midpoint Formula

The midpoint formula is a convenient tool to find the center point between two endpoints. In this exercise, it helps us find the center of the circle, which is crucial for determining the circle's equation. The formula is given by:\[ M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \] Where

• \(x_1\) and \(y_1\) are the coordinates of the first endpoint.
• \(x_2\) and \(y_2\) are the coordinates of the second endpoint.

With this formula, you simply average the x-coordinates to find the x-coordinate of the midpoint and do the same with the y-coordinates for the y-coordinate of the midpoint. In our example, we applied it to the endpoints \((-4,5)\) and \((6,-9)\), resulting in the center of the circle at \((1, -2)\). This simplifies the process and ensures that the calculation is easy to follow.

Center-Radius Form

The center-radius form of a circle is an expression that specifies the equation of the circle using its center and radius. This form is expressed as:\[ (x-h)^2 + (y-k)^2 = r^2 \] Where

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

This equation basically describes all the points \((x, y)\) that are a distance \(r\) away from the center \((h, k)\). After finding the center and radius using midpoint and distance formulas respectively, inserting the values into this form is straightforward. For instance, using the calculated center \((1, -2)\) and radius \(\sqrt{74}\), we derive the equation \((x-1)^2 + (y+2)^2 = 74\). This format is not only elegant but also efficient for quick calculations and checks.

Distance Formula

To find the radius of the circle, which is the distance from the center to a point on the circle, we use the distance formula. The formula is invaluable in this scenario for determining how far apart two points in a plane are:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Here,

• \(x_1\) and \(y_1\) are coordinates of one point.
• \(x_2\) and \(y_2\) are coordinates of the second point.

This involves taking the square difference of the x-values and y-values, summing them, then taking the square root. It's applied to find the distance from center \((1, -2)\) to endpoint \((-4, 5)\), as calculated, it results in a radius of \(\sqrt{74}\). This precise calculation ensures the radius fits perfectly into the center-radius formula to build the complete circle equation.