What is the equation for a circle with a diameter having the following endpoints: N(0,1) and Q(18,1)?
a.)(x - 9)^2 + (y - 1)^2 = 81
b.)(x + 2)^2 + (y - 7)^2 = 56
c.)(x)^2 + (y)^2 = 72
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Answer
a.)(x - 9)^2 + (y - 1)^2 = 81
To find the equation of a circle, we need to know its center and radius. In this case, we are given the endpoints of a diameter, which can help us find the center and subsequently the equation of the circle.
First, let's find the midpoint of the diameter using the endpoints given: N(0,1) and Q(18,1). The midpoint formula is:
Midpoint = ( (x1 + x2) / 2 , (y1 + y2) / 2 )
Plugging in the values, we get:
Midpoint = ( (0 + 18) / 2 , (1 + 1) / 2 )
Midpoint = ( 9 , 1 )
The x-coordinate of the midpoint is 9, and the y-coordinate is 1. This gives us the center of the circle.
Next, we can calculate the distance between the center and one of the endpoints to find the radius. The distance formula is:
Distance = sqrt( (x2 - x1)^2 + (y2 - y1)^2 )
Using the center (9, 1) and one endpoint (0, 1), we get:
Distance = sqrt( (0 - 9)^2 + (1 - 1)^2 )
Distance = sqrt( 81 + 0 )
Distance = sqrt( 81 )
Distance = 9
So, the radius of the circle is 9. Now, we can substitute the center and the radius into the general equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the center of the circle, and r represents the radius.
Substituting the values we found, we get:
(x - 9)^2 + (y - 1)^2 = 9^2
(x - 9)^2 + (y - 1)^2 = 81
Therefore, the equation for a circle with the given endpoints is:
a.)(x - 9)^2 + (y - 1)^2 = 81