A man running a race course notes that the sum of the distance 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.

Question

A man running a race course notes that the sum of the distance 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.

Filo · Accepted Answer

Let A and B be the positions of the two flag posts and P(x,y) be the position of the man such thatPA+PB=10.We know that " If a point moves in a plane in such a way that the sum of its distance from two fixed points is constant then the path is an ellipse and this constant value is equal the length of the major axis of the ellipse".Therefore, the path traced by the man is an ellipse.Length of the major axis =10mPoints A and B are the foci. Taking the origin of the coordinate plane as the centre of the ellipse while taking the major axis along the x-axis the ellipse.The equation of then ellipse will be of the form a2x2​+b2y2​=1 where a is the semi-major axis.2a=10⇒ a=5Distance between the foci (2c)=8⇒ c=ae=4Since, c=a2−b2​, we get4=25−b2​⇒16=25−b2⇒b2=25−16=9⇒ b=3So, the equation of the path traced by the man is 25x2​+9y2​=1

Question Text	A man running a race course notes that the sum of the distance 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.
Answer Type	Text solution:1
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A man running a race course notes that the sum of the distance 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.

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