Problem 23 - 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. $$(-5,-7) \text { and }(1,1)$$

Step-by-Step Solution

Answer

The circle's equation is \((x + 2)^2 + (y + 3)^2 = 25\).

Step 1: Find the Midpoint (Center of the Circle)

To find the center of the circle, we calculate the midpoint of the diameter using the midpoint formula: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]Plugging in the coordinates of the endpoints \((-5, -7)\) and \((1, 1)\), we get:\[\left( \frac{-5 + 1}{2}, \frac{-7 + 1}{2} \right) = (-2, -3)\]So, the center of the circle is \((-2, -3)\).

Step 2: Calculate the Radius

The radius is the distance from the center \((-2, -3)\) to either endpoint, say \((-5, -7)\). Use the distance formula:\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substituting the coordinates \((-5, -7)\) and \((-2, -3)\) into the formula:\[\sqrt{(-5 + 2)^2 + (-7 + 3)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]Thus, the radius of the circle is 5.

Step 3: Write the Equation of the Circle

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. With center \((-2, -3)\) and radius \(5\), the equation becomes:\[(x + 2)^2 + (y + 3)^2 = 5^2\]Simplify to:\[(x + 2)^2 + (y + 3)^2 = 25\]

Key Concepts

Midpoint Formula

The midpoint formula helps us find the center point between two coordinates, which is exactly halfway between them. It is particularly useful when dealing with the geometry of a circle, especially when you need to find the center of the circle using the endpoints of its diameter.
The formula for the midpoint of a line segment made by two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

• Midpoint: \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\)

For example, given the points \((-5, -7)\) and \(1, 1)\), calculating the midpoint would proceed as:

• For the x-coordinate: \(\frac{-5 + 1}{2} = -2\)
• For the y-coordinate: \(\frac{-7 + 1}{2} = -3\)

Therefore, the midpoint, and thus the center of the circle, is conveniently found to be \((-2, -3)\). This point is essential as it acts as the central anchor in determining other properties of the circle.

Distance Formula

The distance formula allows us to measure the line length between two points in a coordinate plane. It is instrumental when calculating the radius of a circle when we know the center and a point on the circle, such as one endpoint of the diameter.
Here's the distance formula you would use:

• Distance: \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

So, using the given endpoints for a diameter \((-5, -7)\) and the center \((-2, -3)\), we find the radius by calculating the distance:

• Subtract the x-coordinates: \(\text{-5+2=-3}\)
• Subtract the y-coordinates: \(\text{-7+3=-4}\)
• Calculate the distance: \(\sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)

This confirms that the radius of the circle is 5 units. Accurately determining the radius is crucial because it defines the size of the circle.

Center-Radius Form

The center-radius form of a circle is a simplified equation that expresses the circle in terms of its center and radius.
It provides an easy way to visualize and work with circles. The formula looks like this:

• \((x - h)^2 + (y - k)^2 = r^2\)

where \( (h, k) \) is the center and \( r \) is the radius of the circle.
Applying this to the example, we have the center at \((-2, -3)\) and a radius of 5. Substitute these values into the formula, and you get:

• \( (x + 2)^2 + (y + 3)^2 = 25 \)

This equation describes our specific circle. It implies every point \((x, y)\) on the circle is exactly 5 units away from the central point \((-2, -3)\). When given the center-radius form, it's easy to derive insights about the geometrical attributes of the circle, like its size and exact position on the coordinate plane.