Draw a diagram to illustrate the problem as shown in the figure below.
Euclid is placed at the origin at (0,0).
Apollonius is 12 m north and 9 m east of Euclid, so its coordinate is (9,12).
Pythagoras is at the arbitrary position (x,y) so that is is at distance d from Euclid and 2d from Apollonius.
From the distance formula, obtain
d² = x² + y² (1)
(2d)² = (x-9)² + (y-12)²
or
4d² = (x-9)² + (y-12)² (2)
Substitute (1) into (2).
4(x² + y²) = x² - 18x + 81 + y² - 24y + 144
3x² + 3y² + 18x + 24y = 225
Divide by 3.
x² + 6x + y² + 8y = 75
Create perfect squares.
(x+3)² - 9 + (y+4)² - 16 = 75
(x+3)² + (y+4)² = 10²
Answer:
The path of Pythagoras is a circle of radius 10 m, centered at (-3, -4).
Euclid is placed at the origin at (0,0).
Apollonius is 12 m north and 9 m east of Euclid, so its coordinate is (9,12).
Pythagoras is at the arbitrary position (x,y) so that is is at distance d from Euclid and 2d from Apollonius.
From the distance formula, obtain
d² = x² + y² (1)
(2d)² = (x-9)² + (y-12)²
or
4d² = (x-9)² + (y-12)² (2)
Substitute (1) into (2).
4(x² + y²) = x² - 18x + 81 + y² - 24y + 144
3x² + 3y² + 18x + 24y = 225
Divide by 3.
x² + 6x + y² + 8y = 75
Create perfect squares.
(x+3)² - 9 + (y+4)² - 16 = 75
(x+3)² + (y+4)² = 10²
Answer:
The path of Pythagoras is a circle of radius 10 m, centered at (-3, -4).