Question 7 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.
Class X1 - Maths -Conic Sections Page 264
Answers
Let equation of elliptical path is in the x²/a² + y²/b² = 1
then, SP + S'P = 2a
SS' = 2C
see attachment,
S and S' are two flag and P is the position of man . then, SP + S'P = 10 [ according to question ]
we know, S'P + SP = 2a = 10
a = 5
Let co-ordinate of two flag { foci} = ( ±c , 0)
a/c distance between flag is 8 m
then, 2c = 8 => c = 4
now, c² = a² - b²
4² = 5² - b²
16 = 25 - b² => b² = 9
now,put the values of a and b in equation of ellipse .
x²/a² + y²/b² = 1
x²/5² + y²/9 = 1 [ a = 5 and b² = 9 ]
x²/25 + y²/9 = 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.
