From the figure, it is clear that the path traced by the man is an ellipse having its foci S1(-ae , 0) in the negative direction of X-axis and S2 (ae, 0) in the positive direction x-axis.

Let equation fo ellipse is  x² /a² + y² /b² = 1.............1

Where b² = a² (1 - e² )

Now given that sum of the distances of the man from two flag posts is 10 m.

So PS₁ + PS₂ = 10

=> 2a = 10

=> a = 5

Again given that distance between the flag posts is 8 m.

i.e. distance between foci is 8 m.

So 2ae = 8

=> ae = 4

=> 5*e = 4

=> e = 4/5

Now b² = a² (1-e² )

=> b² = a² - a² e²

=> b² = 5² - 4²

=> b² = 25-16

=> b² = 9

=> b = 3

Put value of a and b in equation 1, weget the requatied path.

So equation of path is x² /25 + y² /9 = 1