Q.A man running a racecourse notes that the sum of the distances from the two flag posts from 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.

The problem describes a point moving such that the sum of its distances from two fixed points (the flag posts) is constant. This is the defining property of an ellipse, where the two flag posts are the foci.

Place the coordinate system with the origin at the midpoint of the two flag posts and the major axis along the $x$-axis. The distance between the foci is $8$ m, so $2c = 8$, giving $c = 4$ m.

The sum of distances from the foci is always $10$ m, which equals $2a$ by definition of an ellipse. Thus $2a = 10$, so $a = 5$ m.

Using the fundamental relation for an ellipse: …

The heart of this problem is recognizing a locus definition. When a point moves so that the sum of its distances from two fixed points remains constant, and that constant exceeds the distance between the fixed points, the path traced is an ellipse. The two fixed points become the foci of the ellipse.

Here the runner is the moving point, the two flag posts are the foci, and the condition "sum of distances is always $10$ m" is precisely the defining property of an ellipse.

Let's build the equation systematically.

Setting up the coordinate system

1. Place the flag posts symmetrically on the $x$-axis.

   Since the posts are $8$ m apart, position them at $F_1(-4, 0)$ and $F_2(4, 0)$. This makes the distance between them $2c = 8$, so $c = 4$.

2. Identify the ellipse parameter $a$.

   For any point $P(x, y)$ on the ellipse, the sum of distances to the foci is constant:

By definition of an ellipse, this sum equals $2a$. Therefore:

3. Find $b$ using the relationship $c^2 = a^2 - b^2$. We have $a = 5$ and $c = 4$, so:

4. Write the standard equation. With $a^2 = 25$ and $b^2 = 9$, the equation of the ellipse is: $\frac{x^2}{25} + \frac{y^2}{9} = 1$ …