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Question 251076: Three tracking stations have detected a earthquake in the local area. The fist stations is located at the origin ion the county map, where each unit of the grid represents one square mile. The second and third stations are located at (8, -8) and (11, 10), respectively. The epicenter of th4e earthquake is 5 miles from the first tracking station, 13 miles from the second tracking station, and 10 miles from the third tracking station. What are the coordinated of the epicenter of the earthquake? Explain your answer.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
If station 1 is at the origin and the epicenter is 5 miles distant, then the epicenter must lie somewhere on a circle centered at the origin with a 5 mile radius. Likewise, the epicenter must lie on a circle that is centered at (8, -8), the location of station 2, with a 13 mile radius. And finally, it must also lie on a circle centered at (11,10) with a radius of 10 miles. These three circles will intersect in a single point and, given accurate data, this point will be the epicenter.
The equation of a circle centered at ) with radius  is:
^2 + (y - k)^2 = r^2)
Substituting, your three equations are:
^2\ +\ (y\ +\ 8)^2\ =\ 169)
^2\ +\ (y\ -\ 10)^2\ =\ 100)
Rearrange the first two:
^2\ +\ (y\ +\ 8)^2\ -\ 169\ =\ 0)
Set the two expressions equal to one another:
^2\ +\ (y\ +\ 8)^2\ -\ 169\ =\ x^2\ +\ y^2\ -\ 25)
Expand the binomials, collect like terms, and eliminate common factors (the details are left as an exercise for the student) to obtain an equation of the line that passes through the two points of intersection of these first two circles.
Repeat the process with the station 1 and station 3 circles to obtain:
Use either substitution or elimination to solve the system of linear equations. The solution set is the ordered pair containing the coordinates of the epicenter.
John
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