Question

Aman running race course note that the sum of distance from the to two flag post from him is always 10 metre and the distance between the flag post is 8m
an equation of post traced by men​

Answer 1

Given

Refer to attachment for graph

A man is running a race course.

Sum of the distances from the two flag posts from him is always 10, which means it is constant

Distance between the flag posts is 8 m

By the given question the equation of the posts traced by the man is ellipse.

Here the coordinates of the flag posts are foci

Coordinates of the center = ( 0, 0 )

Distance between the flag posts = 8 m

Nothing but distance between foci = 8 m

SS' = 8 m

SC + CS' = 8 m

2a = 8

a = 8/2 = 4

Coordinates of the foci ( ± a, 0 ) = ( ± 4, 0 )

S = ( 4, 0 )

S' = ( - 4, 0 )

Sum of the distance from the flag posts from him = 10 m

PS + PS' = 10

√[ ( x - 4 )² + y² ] + √[ ( x + 4 )² + y² ] = 10

√[ x² + y² - 8x + 16 ] + √[ x² + y² + 8x + 16 ] = 10

√[ x² + y² - 8x + 16 ] = 10 - √[ x² + y² + 8x + 16 ]

Squaring on both sides

x² + y² - 8x + 16 = 10² + x² + y² + 8x + 16 - 20√[ x² + y² + 8x + 16 )

- 16x - 100 = - 20√[ x² + y² + 8x + 16 ]

4x + 25 = 5√[ x² + y² + 8x + 16 ]

( 4x + 25 )² = 25( x² + y² + 8x + 16 )

16x² + 625 + 200x = 25x² + 200x + 5²y² + 400

625 - 400 = 3²x² + 5²y²

225 = 3²x² + 5²y²

15² = 3²x² + 5²y²

1 = 3²x²/15² + 5²y²/15²

1 = x²/5² +y²/3²

x²/25 + y²/9 = 1

Hence the equation of posts traced by the man is x²/25 + y²/9 = 1.