SOLUTION: Equation of a Circle 6. Determine an equation for the circle that has a diameter with endpoints A(-4, 3) and B(4, -3). Thank you!!!!

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Question 189688: Equation of a Circle
6. Determine an equation for the circle that has a diameter with endpoints
A(-4, 3) and B(4, -3).
Thank you!!!!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Step 1) Find the midpoint of the segment with the endpoints A(-4, 3) and B(4, -3).


Remember, the diameter of any circle passes through the center of the circle and the midpoint of the diameter is the same point as the center of the circle.




To find the midpoint, first we need to find the individual coordinates of the midpoint.


X-Coordinate of the Midpoint:




To find the x-coordinate of the midpoint, simply average the two x-coordinates of the given points by adding them up and dividing that result by 2 like this:


x%5Bmid%5D=%28-4%2B4%29%2F2=0%2F2=0


So the x-coordinate of the midpoint is x=0


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Y-Coordinate of the Midpoint:




To find the y-coordinate of the midpoint, simply average the two y-coordinates of the given points by adding them up and dividing that result by 2 like this:


y%5Bmid%5D=%283%2B-3%29%2F2=0%2F2=0


So the y-coordinate of the midpoint is y=0


So the midpoint between the points and is


This means that the center of the circle is the point (0,0)


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Step 2) Find the length of the diameter using the distance formula



d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 Start with the distance formula.


d=sqrt%28%28-4-4%29%5E2%2B%283--3%29%5E2%29 Plug in x%5B1%5D=-4, x%5B2%5D=4, y%5B1%5D=3, and y%5B2%5D=-3.


d=sqrt%28%28-8%29%5E2%2B%283--3%29%5E2%29 Subtract 4 from -4 to get -8.


d=sqrt%28%28-8%29%5E2%2B%286%29%5E2%29 Subtract -3 from 3 to get 6.


d=sqrt%2864%2B%286%29%5E2%29 Square -8 to get 64.


d=sqrt%2864%2B36%29 Square 6 to get 36.


d=sqrt%28100%29 Add 64 to 36 to get 100.


d=10 Take the square root of 100 to get 10.


So our answer is d=10


So the distance between the two points is 10 units.


This means that the length of the diameter is 10 units. Take half of this length to get 5 units.


So the radius of the circle is 5 units



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So we've found the center to be (0,0) and the radius to be 5 units. This means that h=0, k=0, and r=5



%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2 Start with the general equation of a circle


%28x-0%29%5E2%2B%28y-0%29%5E2=5%5E2 Plug in h=0, k=0, and r=5


x%5E2%2By%5E2=5%5E2 Simplify


x%5E2%2By%5E2=25 Square 5 to get 25


So the equation of the circle that has a diameter with endpoints
A(-4, 3) and B(4, -3) is x%5E2%2By%5E2=25


Here's some visual confirmation


Graph of x%5E2%2By%5E2=25