Question

the locations of three receiving stations and the distances to the epicenter of an earthquake are contained in the following three equations: $(x + 3)^{2}+(y - 3)^{2}=4$, $(x + 1)^{2}+(y + 2)^{2}=25$ and $(x + 1)^{2}+(y - 5)^{2}=4$. determine the location of the epicenter.
find the coordinates of the epicenter.
$square$ (type an ordered pair.)

Explanation:

Step1: Expand the equations

The general form of a circle equation is \((x - a)^2+(y - b)^2=r^2\), where \((a,b)\) is the center and \(r\) is the radius.
For \((x + 3)^2+(y - 3)^2 = 4\), expand it: \(x^{2}+6x + 9+y^{2}-6y+9 = 4\), i.e., \(x^{2}+y^{2}+6x-6y+14 = 0\).
For \((x + 1)^2+(y + 2)^2 = 25\), expand it: \(x^{2}+2x + 1+y^{2}+4y+4 = 25\), i.e., \(x^{2}+y^{2}+2x+4y-20 = 0\).
For \((x + 1)^2+(y - 5)^2 = 4\), expand it: \(x^{2}+2x + 1+y^{2}-10y+25 = 4\), i.e., \(x^{2}+y^{2}+2x-10y+22 = 0\).

Step2: Subtract equations to eliminate \(x^{2}\) and \(y^{2}\)

Subtract the second - expanded equation from the first:
\((x^{2}+y^{2}+6x-6y + 14)-(x^{2}+y^{2}+2x+4y-20)=0\)
\(4x-10y+34 = 0\), simplify to \(2x-5y=-17\) (Equation 1).
Subtract the third - expanded equation from the second:
\((x^{2}+y^{2}+2x+4y-20)-(x^{2}+y^{2}+2x-10y+22)=0\)
\(14y-42 = 0\), solve for \(y\): \(y = 3\).

Step3: Substitute \(y\) into Equation 1

Substitute \(y = 3\) into \(2x-5y=-17\).
\(2x-5\times3=-17\), \(2x-15=-17\), \(2x=-2\), \(x=-1\).

Answer:

\((-1,3)\)