Question

A radio antenna is kept perpendicular to the ground by three wires of equal length. The wires touch the ground at three points on a circle whose center is at the base of the antenna. If the wires touch the ground at (9,-19),(-21,-19), and(14,16), what are the coordinates of the base of the antenna?

Answer 1

The coordinates of the base of the antenna is (-6. 1)

Explanation:

Here we are required to find the center of a circle given points on the circumference

The equation of a circle is

(x - h)² + (y - k)² = r²

Where:

x and y are points on the circumference

h and k are coordinates of the center of the circle and

r = The radius of the circle

Since we have the values for the points on the circumference, we are left with three unknowns, which we can find with three equations as follows;

By plugging in the values for x and y at the respective points we get,

At (9, -19) → (9 - h)² + (-19 - k)² = r²...............(1)

At (-21, -19) → (-21 - h)² + (-19 - k)² = r².....,...(2)

At (14, 16 ) → (14 - h)² + (16 - k)² = r² ............(3)

Solving, we get from (1) (-19 - k)² = r² - (9 - h)²

Plugging in the value of (-19 - k)² in equation (2) we get

(-21 - h)² + r² - (9 - h)² = r²

So that (-21 - h)² - (9 - h)² = r² - r² = 0

and (-21 - h)² - (9 - h)² = 0 gives

60·h +360 = 0 or h = -6

Plugging in the value of h = -6 in equation (3) we get

(14 - (-6))² + (16 - k)² = r²

20² + (16 - k)² = r² .................(4)

similarly from equation (1) we get

(9 - (-6))² + (-19 - k)² = r²

15² + (-19 - k)² = r² ................(5)

Subtracting equation (5) from (4) gives

20² - 15² + (16 - k)² - (-19 - k)² = 0

Which gives

-70·k + 70 = 0

or k = 1

Therefore the coordinates of the base of the antenna = (-6. 1).