Which is the equation of a circle with diameter AB with A(5, 4) and B(-1, -4)?
To find the equation of a circle given the diameter AB with A(5, 4) and B(-1, -4), we can use the midpoint formula to find the center of the circle and then use the distance formula to calculate the radius.
Step 1: Find the midpoint of AB.
The midpoint formula is:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Let's apply the formula:
Midpoint = ((5 + (-1))/2, (4 + (-4))/2)
= (4/2, 0/2)
= (2, 0)
The coordinates of the midpoint are (2, 0). This is the center of the circle.
Step 2: Find the radius of the circle.
The distance formula is:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the distance between A(5, 4) and the center (2, 0):
Distance = √((2 - 5)^2 + (0 - 4)^2)
= √((-3)^2 + (-4)^2)
= √(9 + 16)
= √25
= 5
The distance between A and the center is 5 units. This is the radius of the circle.
Step 3: Write the equation of the circle.
The equation of a circle is of the form:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
Using the center (2, 0) and the radius 5, we have:
(x - 2)^2 + (y - 0)^2 = 5^2
Simplifying the equation, we get:
(x - 2)^2 + y^2 = 25
So, the equation of the circle with diameter AB is (x - 2)^2 + y^2 = 25.