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# Epicenter

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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 A(2,3)A(2,3), 4 miles away from B(−5,3)B(−5,3), and 5 miles away from C(−1,−2)C(−1,−2).

Feb 22, 2019
edited by Guest  Feb 22, 2019

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We have the equations of these three circles

(x - 2)^2 + ( y - 3)^2 = 9      ⇒  (y - 3)^2 = 9 - (x - 2)^2    (1)

(x + 5)^2 + (y - 3)^2 = 16          (2)

(x + 1)^2 + ( x + 2)^2 = 25

Sub (1) into (2) for (y - 3)^2.....and we have that

( x + 5)^2 + 9 - (x - 2)^2    = 16   simplify

x^2 + 10x + 25 + 9 - x^2 + 4x - 4  = 16

14x + 30 = 16

14x = - 14

x = -1   this is the x coordinate of the epicenter

Use (1) to find the y coordinate of the epicenter

( y - 3)^2 =   9 - (-1 - 2)^2

(y - 3)^2 = 9 - 9

(y - 3)^2 = 0     take the square root

y - 3 = 0

y = 3

The epicenter is  (-1,3)

This graph shows the situation :  https://www.desmos.com/calculator/6ofwemeyej   Feb 22, 2019