Euclid and apollonius stand on two seperate portions which apollonius 12 meters due worth and 9 meters due east of euclid. meanwhile, phytagoras is jogging in a path in such away that his distance from apollonius is also always twice that from euclid. show that phytagoras' path is circular.

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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

The path of Pythagoras is a circle of radius 10 m, centered at (-3, -4).

The first option would be correct if it were one half and not one