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.
Given, the distance between two flag post is 8 m and the sum of the distances from two flag posts from the man is always 10.
Let, A and B be the position of two flag posts and P ( x , y ) be the position of the man.
PA + PB = 10 .
Point moves in a plane such that sum of the distances from two fixed points is constant. Therefore, the path is an ellipse and constant value is the length of major axis,
2 a = 10 a = 5
Since, distance between two flag post is 8,
2 c = 8 c = 4
The equation of ellipse is represented as,
x 2 a 2 + y 2 b 2 = 1 (1)
Where c = a 2 − b 2
Substitute the value of a and c in c = a 2 − b 2 .
c = a 2 − b 2 4 = 5 2 − b 2 b 2 = 25 − 16 b = ± 9
Substitute values of a and b in (1),
x 2 5 2 + y 2 3 2 = 1 x 2 25 + y 2 9 = 1
Thus, the equation of the locus is x 2 25 + y 2 9 = 1 .
Given, the distance between two flag post is
Let, A and B be the position of two flag posts and
Point moves in a plane such that sum of the distances from two fixed points is constant. Therefore, the path is an ellipse and constant value is the length of major axis,
Since, distance between two flag post is 8,
The equation of ellipse is represented as,
Where
Substitute the value of
Substitute values of
Thus, the equation of the locus is
