Answer:
The coordinates of the base of the antenna is (-6. 1)
Step-by-step 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).
