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

Step-by-Step Solution

Answer

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

Step 1: Find the Midpoint

To find the midpoint of a line segment given endpoints \((-5,-7)\) and \(1,1\), use the midpoint formula: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Substitute the given values: \[ \left( \frac{-5 + 1}{2}, \frac{-7 + 1}{2} \right) = \left( \frac{-4}{2}, \frac{-6}{2} \right) = (-2, -3) \]. Thus, the midpoint (and center of the circle) is \((-2,-3)\).

Step 2: Calculate the Radius

To find the radius, first calculate the distance between the center \((-2, -3)\) and one of the endpoints, say \(1, 1\). Use the distance formula: \[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. Substitute the values: \[ \sqrt{(1 - (-2))^2 + (1 - (-3))^2} = \sqrt{(1 + 2)^2 + (1 + 3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]. The radius is 5.

Step 3: Write the Equation in Center-Radius Form

The standard form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius. We have the center \((-2, -3)\) and radius \5\. Substitute these into the equation:\((x + 2)^2 + (y + 3)^2 = 5^2\). Simplify the equation: \( (x + 2)^2 + (y + 3)^2 = 25 \). This is the equation of the circle.

Key Concepts

Midpoint Formula

The midpoint formula is a handy tool for finding the middle point on a line segment between two points. If you have two endpoints of a diameter, like

• Point A: \((-5, -7)\)
• Point B: \((1, 1)\)

You use the formula: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]This involves adding the x-coordinates and y-coordinates separately, then dividing each by two:

• First, \( \frac{-5 + 1}{2} = -2 \)
• Then, \( \frac{-7 + 1}{2} = -3 \)

Thus, the midpoint is \((-2, -3)\). This point is also the center of the circle when dealing with a circle's diameter. Having the midpoint is crucial because it serves as our circle's center for writing the circle's equation.

Radius Calculation

Once the center of the circle is known, the next step is determining the radius, which is the distance from this center to any endpoint of the diameter. In our example, the center is at \((-2, -3)\), and we choose endpoint \((1, 1)\). To find this distance, we use the distance formula, which is:\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]For our particular points:

• Calculate \((1 - (-2)) = 1 + 2 = 3\)
• Calculate \((1 - (-3)) = 1 + 3 = 4\)

Thus, the distance is:\[\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]This 5-unit distance is the radius of our circle, showing how far the boundary of the circle extends from its center.

Distance Formula

The distance formula determines the straight-line distance between two points in a plane. It is vital in geometry, especially when calculating the size of geometric figures like circles. Given two points

• \((x_1, y_1)\)
• \((x_2, y_2)\)

the distance formula is:\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula works by creating a right triangle from the two points, with differences in x and y as the triangle's legs. The distance is then the hypotenuse. In our circle context, it confirms that the radius is 5 units. The distance formula is a general tool useful beyond circles, aiding any scenario requiring precise point-to-point distance measurement.

Center-Radius Form

The center-radius form is the standard way to express the equation of a circle. It emphasizes the circle's center coordinates and its radius, offering a straightforward description of the circle's size and location. The general form is:\[(x - h)^2 + (y - k)^2 = r^2\]

• \((h, k)\) represents the circle's center coordinates
• \(r\) stands for the circle's radius

For our specific circle:

• Center \((-2, -3)\)
• Radius \(5\)

The equation becomes:\[(x + 2)^2 + (y + 3)^2 = 25\]This equation represents all points that lie on the circle, each spaced exactly 5 units from the center. This form is easy to interpret and helps in visualizing or sketching the circle's position on a coordinate grid.