Question

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

Answer 1

The coordinates of the epicenter are (1,2) ie x=1, y =2

Explanation:

We use the fact that the distance d between 2 points, (C y₁) and (x₂, y₂) is computed by the Pythagorean formula:
(x₁ - x₂)² + (y₁ - y₂) = d²

Let the epicenter be located at (x, y)

Distance dA from point A(5,2) is known to be 4 and is given by the equation:
(x-5)² + (y - 2)² = dA² = 4² = 16 ==>

(x-5)² + (y-2)² = 16 (1)

Similarly, the distance from point B(-2,2) is given as 3. Therefore,
(x - (-2))² + (y-2)² = 3² = 9 or
(x + 2)² + (y-2)² = 9 (2)

Distance from point C(1,-3) ==>

(x-1)² + (y-(-3))² ==> (x-1)² + (y +3)² = 25 ...(3)

(1) - (2) cancels out the y terms and gives us
(x-5)² - (x+2)² = 7 (4)

This simplifies to
x² + -10x + 25 - (x² +4x + 4) = 7
or
x² + -10x + 25 -x² -4x - 4 = 7
-10x -4x = 7 - 21 = -14
Therefore, -14x = -14 giving us x = 1

Substituting for x = 1 in equation (3) gives us
(1-1)² + (y+3)² = 25 or
y + 3 = √25 = 5 giving y = 2

So we get the epicenter coordinates as x=1, y =2

Crosscheck in each of the equations (1), (2) and (3) for consistency
Equation 1 : (1-5)² + (2-2)² = -4² = 16 Check!

Equation 2: (1-(-2))² + (2-2)² = (1+2)² = 3² = 9 Check!

Equation 3: (1-1)² + 2-(-3))² = (2+3)² = 5² Check!