To find the equation of a circle based on the diameter defined by points A and B, we start with the following steps:
Step 1: Find the midpoint (center of the circle)
The midpoint formula is given by:
\[ (x,y) = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \]
Let \(A(-4, 2)\) and \(B(-4, 16)\):
\[ h = \frac{-4 + -4}{2} = \frac{-8}{2} = -4 \] \[ k = \frac{2 + 16}{2} = \frac{18}{2} = 9 \]
So, the center of the circle is \((-4, 9)\).
Step 2: Find the distance (diameter of the circle)
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, let \(x_1 = -4\), \(y_1 = 2\), \(x_2 = -4\), \(y_2 = 16\):
\[ d = \sqrt{((-4) - (-4))^2 + (16 - 2)^2} \]
This simplifies to:
\[ d = \sqrt{(0)^2 + (14)^2} = \sqrt{196} = 14 \]
So, the measure of the diameter of the circle is \(14\).
Step 3: Find the radius of the circle
The radius is half of the diameter:
\[ r = \frac{d}{2} = \frac{14}{2} = 7 \]
Step 4: Write the equation of the circle
The general equation of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \(h = -4\), \(k = 9\), and \(r = 7\) into the equation:
\[ (x - (-4))^2 + (y - 9)^2 = 7^2 \] \[ (x + 4)^2 + (y - 9)^2 = 49 \]
Final Answers
- Measure of the diameter of the circle: \(14\)
- Measure of the radius of the circle: \(7\)
- Center of the circle (h, k): \((-4, 9)\)
- Equation of the circle: \((x + 4)^2 + (y - 9)^2 = 49\)