Misc 7(Method 1)
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.
Let man be standing on point P(x, y)
There are two flag posts S & S
Given
PS + PS = 10
& SS = 8 m
Let S & S be on x-axis
such that Origin (O) be the mid-point of S S
So, OS = OS = 4 m
So, coordinates of S = S(4, 0)
coordinates of S = S ( 4, 0)
We know that
distance between two points (x1, y1) & (x2, y2)
d = 2 1 2 + 2 1 2
Now, finding PS & PS
PS = ( 4) 2 + 0 2
PS = +4 2 + 2
and
PS = 4 2 + 0 2
PS = 4 2 + 2
Now,
PS + PS = 10
Putting values
4 2 + 2 + +4 2 + 2 = 10
4 2 + 2 = 10 +4 2 + 2
Squaring both sides
4 2 + 2 2 = 10 +4 2 + 2 2
4 2 + 2 = 102 + +4 2 + 2 2 20 +4 2 + 2
4 2 + 2 = 100 + +4 2 + 2 20 +4 2 + 2
x2 + 42 8x + 2 = 100 + x2 + 42 + 8x + 2 20 +4 2 + 2
x2 + 42 8x + 2 100 x2 42 8x 2 = 20 +4 2 + 2
16x 100 = 20 +4 2 + 2
20 +4 2 + 2 = 16x + 100
20 +4 2 + 2 = 4(4x + 25)
5 +4 2 + 2 = (4x + 25)
Squaring both sides
5 +4 2 + 2 2 = (4x + 25)2
25 +4 2 + 2 = (4x + 25)2
25(x2 + 42 + 8x + y2) = (4x)2 + (25)2 + 2(4x)(25)
25x2 + 400 + 200x + 25y2 = 16x2 + 625 + 200x
9x2 + 25y2 = 225
9 2 225 + 25 2 225 = 1
+ = 1
Misc 7(Method 2)
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 S & S be two flag past
The sum of distance of man P
from S & S is equal to 10
Since the sum of distance of a point from any two fixed points S & S in the plane is constant,
it forms on ellipse.
So, S & S are the foci of the ellipse
x-axis is the major axis, y-axis is the minor axis
Let equation of ellipse be 2 2 + 2 2 = 1
Now, given that
The sum of distance of man P from S & S = 10
i.e. Sum of distance of a point from foci is = 10
2a = 10 m
a = 10 2 = 5m
Also it is given that
Distance between flag posts SS = 8m
Distance between two foci = 8 m
2c = 8
c = 8 2 = 4
Now we need to find b
We know that
c2 = a2 b2
42 = 52 b2
16 = 25 b2
b2 = 25 16
b2 = 9
b = 3
Now equation of ellipse is
2 2 + 2 2 = 1
2 5 2 + 2 3 2 = 1
+ = 1
Thus, the required equation is 2 25 + 2 9 = 1

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.