Problem 92 - College Algebra and Calculus: An Applied Approach

Question

Find the standard form of the equation of the specified circle. Endpoints of a diameter: \((-4,-1),(4,1)\)

Step-by-Step Solution

Answer

The standard form of the equation of the circle with endpoints of a diameter at (-4,-1) and (4,1) is \(x^2 + y^2 = 17\).

Step 1: Calculate the Center

The center point of a circle given two endpoints of a diameter is just the midpoint of the two endpoints. The midpoint formula is \((x_1 + x_2) / 2, (y_1 + y_2) / 2\). Substituting the given points, we find the midpoint: \(((-4+4)/2 , (-1+1)/2) = (0,0)\). So, the center of the circle (h, k) is (0, 0).

Step 2: Calculate the Radius

The radius is half the length of the diameter. The formula for the length of a diameter (d) given the coordinates of its two endpoints is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Using this formula, the length of the diameter becomes: \(\sqrt{(4 - (-4))^2 + (1 - (-1))^2} = \sqrt{(8)^2 + (2)^2} = \sqrt{64 + 4} = \sqrt{68}\). Therefore, the radius r is \(d/2 = \sqrt{68}/2 = \sqrt{17}\).

Step 3: Write the Equation in Standard Form

Substitute the center (h, k) and radius r into the standard equation \((x - h)^2 + (y - k)^2 = r^2\). Therefore the standard form of the equation of the circle is \((x-0)^2 + (y-0)^2 = (\sqrt{17})^2\). After simplification, this becomes: \(x^2 + y^2 = 17\).

Key Concepts

Midpoint Formula

The midpoint of a line segment is exactly halfway between the two endpoints. It's like finding the balance point on a seesaw. To find the midpoint, you use this simple formula: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). This formula averages the x-coordinates and the y-coordinates of the endpoints.

For example, with endpoints \((-4,-1)\) and \((4,1)\), the midpoint would be:

• \(x\)-coordinate: \(\frac{-4 + 4}{2} = 0\)
• \(y\)-coordinate: \(\frac{-1 + 1}{2} = 0\)

Therefore, the midpoint, which also serves as the center of our circle, is \((0,0)\). This process helps us identify the precise center point when given diameter endpoints.

Radius Calculation

Once you have the center of a circle, the next step is finding the radius. The radius is half the diameter—imagine slicing a pizza in half. You find the diameter's length using the distance formula, \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). This tells you how far apart the endpoints are.

Using our endpoints \((-4,-1)\) and \((4,1)\), the diameter length becomes:

• \(\sqrt{(4 - (-4))^2 + (1 - (-1))^2} = \sqrt{8^2 + 2^2} = \sqrt{68}\)

So, the radius is half of this length:

• \( r = \frac{\sqrt{68}}{2} = \sqrt{17} \)

This gives us the distance from the circle's center to any point on the circle.

Diameter Endpoints

When given the endpoints of a diameter, you gain valuable information about the circle. These points are directly opposite each other across the center.

Understanding the relationship:

• The midpoint of the endpoints is the circle's center.
• The distance between endpoints, calculated using the distance formula, is the diameter.

In our exercise, the endpoints \((-4,-1)\) and \((4,1)\) provide us:

• A center at \((0,0)\), calculated using the midpoint formula
• A diameter length of \(\sqrt{68}\)

With these components, calculating the radius and formulating the circle's equation becomes straightforward. Recognizing these connections simplifies the circle equation process!