A man running a racecourse notes that the sum of the distances from the two flag posts form him is always 10 m and the distance between the flag posts is 8 m. find the equation of the posts traced by the man.

Let A and B be the positions of the two flag posts and P(x, y) be the position of the man.

Accordingly, P A + P B = 10.

We know that if a point moves in a plane such that the sum of its distances from two fixed points remains constant, the path traced is an ellipse, and this constant value equals the length of the major axis.

Therefore, the path described by the man is an ellipse where the length of the major axis is 10 m, while points A and B are the foci.

Taking the origin of the coordinate plane as the centre of the ellipse, with the major axis along the x-axis, the equation of the ellipse is:

(x² / a²) + (y² / b²) = 1, where a is the semi-major axis.

Accordingly, 2a = 10 ⇒ a = 5.

Distance between the foci (2c) = 8 ⇒ c = 4.

Using the relation c = √(a² − b²), we get:

4 = √(25 − b²)

⇒ 16 = 25 − b²

⇒ b² = 9 ⇒ b = 3.

Thus, the equation of the path traced by the man is:

(x² / 25) + (y² / 9) = 1.