Problem 5 - Algebra 2

Question

Write an equation for the circle that satisfies each set of conditions. endpoints of a diameter at \((-4,1)\) and \((4,-5)\)

Step-by-Step Solution

Answer

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

Step 1: Find the Center of the Circle

The center of the circle is the midpoint of the diameter. Use the midpoint formula, which is \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]where \((x_1, y_1) = (-4, 1)\) and \((x_2, y_2) = (4, -5)\). Substitute these values to find the midpoint:\[\left( \frac{-4 + 4}{2}, \frac{1 + (-5)}{2} \right) = (0, -2)\]Thus, the center of the circle is \((0, -2)\).

Step 2: Find the Radius of the Circle

The radius is half the length of the diameter. First, calculate the length of the diameter using the distance formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]where \((x_1, y_1) = (-4, 1)\)and\((x_2, y_2) = (4, -5)\). Substitute these values:\[d = \sqrt{(4 - (-4))^2 + (-5 - 1)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10\]The radius is half of this, so\(r = \frac{10}{2} = 5\).

Step 3: Write the Equation of the Circle

The general equation of a circle is \[(x-h)^2 + (y-k)^2 = r^2\]where \((h, k)\) is the center of the circle and \(r\) is the radius. From previous steps, \((h, k) = (0, -2)\)and \(r = 5\). Substitute these values into the equation:\[(x-0)^2 + (y+2)^2 = 5^2 \]Simplify to get:\[ x^2 + (y+2)^2 = 25 \]This is the equation of the circle.

Key Concepts

Midpoint Formula

To find the center of a circle when you have the endpoints of its diameter, you can use the midpoint formula. This formula helps in identifying the point exactly halfway between two other points. The formula is given by \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]where - \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the endpoints.When dealing with endpoints \((-4, 1)\) and \((4, -5)\), substitute these into the formula:- The x-coordinate: \(\frac{-4 + 4}{2}\) = 0.- The y-coordinate: \(\frac{1 + (-5)}{2}\) = -2.So, the midpoint (center of this circle) is at \((0, -2)\). This center is crucial in further computations such as finding the radius.

Distance Formula

The distance formula is used to compute the length between two points in a plane. It is derived from the Pythagorean theorem and is particularly useful for finding the length of a line segment connecting two points. The formula is:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]In our case, to calculate the length of the diameter of the circle when the points are \((-4, 1)\) and \((4, -5)\), we perform the following calculation:- Substitute the coordinates: \[\sqrt{(4 - (-4))^2 + (-5 - 1)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10\]This calculation reveals the diameter is 10 units. Knowing this is pivotal in finding the circle's radius.

Radius of a Circle

The radius of a circle is a line segment from the center of the circle to any point on its circumference. When given a diameter, finding the radius is straightforward:- The radius is simply half the diameter.Given the previously computed diameter of 10, we find:\[r = \frac{10}{2} = 5\]This radius of 5 is essential when writing the full equation of the circle. The radius is not just a measure of size, it's part of the definition of a circle's equation.

Equation of a Circle

Having the center and radius allows us to write the equation of the circle. The standard form of a circle’s equation is:\[(x-h)^2 + (y-k)^2 = r^2\]where - \((h, k)\) is the center,- \(r\) is the radius of the circle.For our circle with a center at \((0, -2)\) and a radius of 5, we substitute:- \((h, k) = (0, -2)\)- \(r=5\)The equation becomes:\[(x-0)^2 + (y+2)^2 = 5^2\]Simplifying further:\[x^2 + (y+2)^2 = 25\]This is your circle's equation. Understanding and applying these steps provides a solid basis for solving circle-related problems effectively.