Question

Three cell phones towers can be modeled by the points X(6,0), Y(8,4), and Z(3,9). Determine the location of another cell phone tower equidistant from the other three, and write an equation for the circle.

Answer 1

The location of the new cell phone tower is
`(h,k) = (3,4)`, and the equation of the circle is
`x^(2)+y^(2) -6\cdot x - 8\cdot y = 0`.

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:

`x^(2) + y^(2)+A\cdot x + B\cdot y +C = 0` (1)

Where:

`x` - Independent variable.

`y` - Dependent variable.

`A`,
`B`,
`C` - Circunference constants.

Given the number of variable, we need the location of three distinct points:

`(x_(1),y_(1)) = (6,0)`

`36 +6\cdot A + C = 0`

`(x_(2),y_(2)) = (8,4)`

`80 + 8\cdot A + 4\cdot B + C = 0`

`(x_(3),y_(3)) = (3,9)`

`90 + 3\cdot A + 9\cdot B + C = 0`

Then, we have the following system of linear equations:

`6\cdot A + C = -36` (2)

`8\cdot A +4\cdot B + C = -80` (3)

`3\cdot A + 9\cdot B + C = -90` (4)

The solution of this system is:

`A = -6`,
`B = -8`,
`C = 0`

By comparing the general form with the standard form of the equation of the circunference is:

`A = -2\cdot h` (5)

`B = -2\cdot k` (6)

`C = h^(2)+k^(2)-r^(2)` (7)

Where:

`h`,
`k` - Coordinates of the center of the circle.

`r` - Radius of the circle.

If we know that
`A = -6`,
`B = -8` and
`C = 0`, then coordinates of the center of the circle and its radius are, respectively:

`h = -(A)/(2)`

`k = -(B)/(2)`

`r = \sqrt{h^(2)+k^(2)-C}`

`h = 3`,
`k = 4`,
`r = 5`

The location of the new cell phone tower is
`(h,k) = (3,4)`, and the equation of the circle is
`x^(2)+y^(2) -6\cdot x - 8\cdot y = 0`.