Question

Find the standard form of the equation of each circle. With endpoints of a diameter at (1,4) and (-3,2)

Short answer

(x + 1)^2 + (y - 3)^2 = 5.

Step-by-step solution

Find the Midpoint

To find the center of the circle, calculate the midpoint of the diameter. Use the midpoint formula: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). \)Plugging in our points (1, 4) and (-3, 2): \( \left( \frac{1 + (-3)}{2}, \frac{4 + 2}{2} \right) = (-1, 3). \)

Calculate the Radius

To find the radius, calculate the distance between the midpoint (center) and one of the endpoints of the diameter. Use the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)Substituting the midpoint (-1, 3) and an endpoint (1, 4): \( d = \sqrt{(1 + 1)^2 + (4 - 3)^2} \) \( d = \sqrt{2^2 + 1^2} \) \( d = \sqrt{4 + 1} = \sqrt{5}. \)Therefore, the radius is \( \sqrt{5} \).

Write the Equation in Standard 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.Substitute the values found: \( (h, k) = (-1, 3) \) and r = \( \sqrt{5} \).The resulting equation is \( (x + 1)^2 + (y - 3)^2 = 5. \)

Key concepts

Midpoint Formula

To determine the center of the circle from its diameter, we use the midpoint formula. The midpoint formula is a way of finding the center point between two points in a coordinate plane.
It’s given by: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).

Here's how it works:

• Step 1: Add the x-coordinates of your two points together.
• Step 2: Add the y-coordinates of your two points together.
• Step 3: Divide both sums by 2.

Applying this to our given points (1, 4) and (-3, 2), we have: \[ \left( \frac{1 + (-3)}{2}, \frac{4 + 2}{2} \right) = (-1, 3) \].

This point, (-1, 3), is the midpoint of the diameter and thus the center of the circle.

Distance Formula

Once we have the center of the circle, the next step is determining the radius. The radius is the distance between the center of the circle and any point on the perimeter (in this case, one of the endpoints of the diameter).
To find this distance, we use the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

The steps to use the distance formula are as follows:

• Step 1: Subtract the x-coordinates of the two points.
• Step 2: Subtract the y-coordinates of the two points.
• Step 3: Square both results from steps 1 and 2.
• Step 4: Add the squares.
• Step 5: Take the square root of the sum.

Applying this to our points (the midpoint (-1, 3) and one endpoint (1, 4)): \[ d = \sqrt{(1 + 1)^2 + (4 - 3)^2} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \].

Thus, the radius of the circle is \( \sqrt{5} \).

Standard Form of a Circle

Finally, we formulate the equation of the circle in its standard form using the center and radius calculated in the previous steps.
The standard form of the equation of a circle is given by: \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \( r \) is the radius.

Here's a quick summary of how to use the standard form:

• Step 1: Identify your center point \((h, k)\).
• Step 2: Determine your radius \( r \).
• Step 3: Plug these values into the standard equation format.

For our exercise, the center is (-1, 3) and the radius is \( \sqrt{5} \). Substituting these values, the equation becomes: \( (x + 1)^2 + (y - 3)^2 = 5 \).
This is the standard form of the equation for our circle. Remember, the standard form is useful because it clearly shows the circle's center and radius.