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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo

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