The epicenter of an earthquake is the point on Earth’s surface directly above the earthquake’s origin. A seismograph can be used to determine the distance to the epicenter of an earthquake. Seismographs are needed in three different places to locate an earthquake’s epicenter. Find the location of the earthquake’s epicenter if it is 3 miles away from (2,1) , 4 miles away from (5,-3) , and 5 miles away from (10,1)

An epicenter can be defined as the point on the surface of planet Earth, which is perpendicular to the hypocenter (focus) and directly above the earthquake’s origin. Also, it indicates where the seismic waves from an earthquake are most intense.

For this exercise, we would set the epicenter at x and y. Using the equation of a circle, we have the following:

At A (2, 1) and 3 miles away;

[tex](y-1)^2+(x-2)^2=3^2[/tex] ......equation 1.

At B (-5, 3) and 4 miles away;

[tex](y-3)^2+(x-(-5))^2=4^2\\\\(y-3)^2+(x+5)^2=4^2[/tex] ......equation 2.

At C (10, 1) and 5 miles away;

[tex](y-1)^2+(x-10)^2=5^2[/tex] ......equation 3.

Subtracting eqn. 2 from eqn. 1, we have:

[tex](x-2)^2-[(x+5)^2]=3^2-4^2\\\\x^2-2x-2x+4-[(x^2+5x+5x+25)]=9-16\\\\x^2-4x+4-x^2-10x-25=-7\\\\-14x-21=-7\\\\-14x=-7+21\\\\-14x=14\\\\x=\frac{-14}{14}[/tex]

x = -1 when y = 1.

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