Problem 19 - Algebra 2

Question

Write an equation for the circle that satisfies each set of conditions. endpoints of a diameter at \((11,18)\) and \((-13,-19)\)

Step-by-Step Solution

Answer

The equation of the circle is \((x + 1)^2 + (y + 0.5)^2 = 486.25\).

Step 1: Identify the center of the circle

The center of the circle, which is also the midpoint of the diameter, can be found using the midpoint formula: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]Substitute \((x_1, y_1) = (11, 18)\) and \((x_2, y_2) = (-13, -19)\): \[\left( \frac{11 + (-13)}{2}, \frac{18 + (-19)}{2} \right) = \left( \frac{-2}{2}, \frac{-1}{2} \right) = (-1, -0.5)\]So, the center of the circle is \((-1, -0.5)\).

Step 2: Calculate the radius of the circle

Calculate the distance between one endpoint of the diameter and the center. The distance formula is used: \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Use endpoint \((11, 18)\) and center \((-1, -0.5)\): \[\sqrt{(11 - (-1))^2 + (18 - (-0.5))^2} = \sqrt{(12)^2 + (18.5)^2}\]Calculate:\[= \sqrt{144 + 342.25} = \sqrt{486.25}\]

Step 3: Write the equation of the circle

To write the circle's equation, use the formula:\[(x - h)^2 + (y - k)^2 = r^2\]Where \((h, k)\) is the center and \(r\) is the radius. We have \((-1, -0.5)\) as the center and \(r^2 = 486.25\):\[(x + 1)^2 + (y + 0.5)^2 = 486.25\]

Key Concepts

Midpoint Formula

The midpoint formula is a simple mathematical tool used to find the center point between two endpoints of a line segment. Imagine you have a line with two endpoints, and you want to determine the exact middle point along that line. That's where the midpoint formula comes in handy. It is given by:
\[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]

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

The midpoint provides a crucial piece of information: for a circle, it is the center when you're given the diameter's endpoints. Knowing this, you can then move forward to calculate other elements of the circle, such as its radius.

Distance Formula

The distance formula is essential for finding the length between two points in a plane. It is derived from the Pythagorean theorem and takes into account the difference in both the horizontal and vertical coordinates of the points:
\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula calculates the direct "straight line" distance, which is important for many applications.

• It's useful in determining the circle's radius when you know the center and a point on the circle.
• The formula can also verify that the endpoints of a diameter indeed lie on the circle.

By knowing how to use this formula, you can ensure accuracy in your geometric calculations and confidently find distances between any two points on the Cartesian plane.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. It is directly related to the length of the diameter, as the diameter is simply twice the length of the radius. The radius plays a vital role in defining the circle's size.

• Once the center of a circle is established using the midpoint formula, the radius can be found using the distance formula from the center to any endpoint provided on the circle.
• In equations, the radius often appears as \(r\) or \(r^2\) (with \(r^2\) representing the radius squared).

Understanding the radius is crucial because it allows you to write the equation of the circle accurately. It directly impacts the circle's area, and understanding how to calculate it from endpoints helps in multiple mathematical contexts.

Diameter Endpoints

Diameter endpoints are the two points that lie on a circle and form a line passing through the center, effectively cutting the circle in half. These endpoints are crucial because they help in determining the circle's equation by providing two important pieces of information: location of the center and length of the diameter.

• The midpoint of the diameter is the circle's center when calculating the equation.
• The distance between the endpoints is twice the radius, providing another method to find the radius if the circle's center is known.

By using the diameter endpoints effectively, you can solve intricate problems related to circle geometry. This includes deriving both the circle's equation and understanding the broader relationship between the circle's geometrical properties.