SOLUTION: How do you solve the problem that asks: Find the equation of the circle with the diameter AB, given, that the coordinates of A and B are (-6,1) and (4,-5).

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: How do you solve the problem that asks: Find the equation of the circle with the diameter AB, given, that the coordinates of A and B are (-6,1) and (4,-5).      Log On


   

Question 582101: How do you solve the problem that asks: Find the equation of the circle with the diameter AB, given, that the coordinates of A and B are (-6,1) and (4,-5).
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Recall that the general equation of a circle is %28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2.


So we need the center (h,k) and the radius squared r%5E2.


First, let's find the center (h,k).


Since the center is the midpoint of the line segment with endpoints (-6,1) and (4,-5), we need to find the midpoint.


X-Coordinate of Midpoint = %28x%5B1%5D%2Bx%5B2%5D%29%2F2+=+%28-6%2B4%29%2F2=-2%2F2+=+-1


Since the x coordinate of midpoint is -1, this means that h=-1


Y-Coordinate of Midpoint = %28y%5B1%5D%2By%5B2%5D%29%2F2+=+%281%2B-5%29%2F2=-4%2F2+=+-2


Since the y coordinate of midpoint is -2, this means that k=-2


So the center is the point (-1, -2)


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Now let's find the radius squared


Use the formula r%5E2=%28x-h%29%5E2%2B%28y-k%29%5E2, where (h,k) is the center and (x,y) is an arbitrary point on the circle.


In this case, h=-1 and k=-2. Also, x=-6 and y=1. Plug these values into the equation above and simplify to get:


r%5E2=%28-6--1%29%5E2%2B%281--2%29%5E2


r%5E2=%28-5%29%5E2%2B%283%29%5E2


r%5E2=25%2B9


r%5E2=34


So because h=-1, k=-2, and r%5E2=34, this means that the equation of the circle that passes through the points (-6,1) and (4,-5) (which are the endpoints of the diameter) is %28x%2B1%29%5E2%2B%28y%2B2%29%5E2=34.