Answer:
The location of the new cell phone tower is [tex](h,k) = (3,4)[/tex], and the equation of the circle is [tex]x^{2}+y^{2} -6\cdot x - 8\cdot y = 0[/tex].
Step-by-step explanation:
The location of the cell phone tower coincides with the location of a circunference passing through the three cell phone towers. By Analytical Geometry, the equation of the circle is represented by the following general formula:
[tex]x^{2} + y^{2}+A\cdot x + B\cdot y +C = 0[/tex] (1)
Where:
[tex]x[/tex] - Independent variable.
[tex]y[/tex] - Dependent variable.
[tex]A[/tex], [tex]B[/tex], [tex]C[/tex] - Circunference constants.
Given the number of variable, we need the location of three distinct points:
[tex](x_{1},y_{1}) = (6,0)[/tex]
[tex]36 +6\cdot A + C = 0[/tex]
[tex](x_{2},y_{2}) = (8,4)[/tex]
[tex]80 + 8\cdot A + 4\cdot B + C = 0[/tex]
[tex](x_{3},y_{3}) = (3,9)[/tex]
[tex]90 + 3\cdot A + 9\cdot B + C = 0[/tex]
Then, we have the following system of linear equations:
[tex]6\cdot A + C = -36[/tex] (2)
[tex]8\cdot A +4\cdot B + C = -80[/tex] (3)
[tex]3\cdot A + 9\cdot B + C = -90[/tex] (4)
The solution of this system is:
[tex]A = -6[/tex], [tex]B = -8[/tex], [tex]C = 0[/tex]
By comparing the general form with the standard form of the equation of the circunference is:
[tex]A = -2\cdot h[/tex] (5)
[tex]B = -2\cdot k[/tex] (6)
[tex]C = h^{2}+k^{2}-r^{2}[/tex] (7)
Where:
[tex]h[/tex], [tex]k[/tex] - Coordinates of the center of the circle.
[tex]r[/tex] - Radius of the circle.
If we know that [tex]A = -6[/tex], [tex]B = -8[/tex] and [tex]C = 0[/tex], then coordinates of the center of the circle and its radius are, respectively:
[tex]h = -\frac{A}{2}[/tex]
[tex]k = -\frac{B}{2}[/tex]
[tex]r = \sqrt{h^{2}+k^{2}-C}[/tex]
[tex]h = 3[/tex], [tex]k = 4[/tex], [tex]r = 5[/tex]
The location of the new cell phone tower is [tex](h,k) = (3,4)[/tex], and the equation of the circle is [tex]x^{2}+y^{2} -6\cdot x - 8\cdot y = 0[/tex].
