Q7.    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.

Answers (1)

As we know that if a point moves in a plane in such a way that its distance from two-point remain constant then the path is an ellipse.

Now, According to the question,

the distance between the point from where the sum of  the distance from a point is constant =  10

\Rightarrow 2a=10\Rightarrow a=5

Now, the distance between the foci=8

\Rightarrow 2c=8\Rightarrow c=4

Now, As we know the relation,

c^2=a^2-b^2

b^2=a^2-c^2

b=\sqrt{a^2-c^2}=\sqrt{5^2-4^2}=\sqrt{25-16}=\sqrt{9}=3

Hence the equation of the ellipse is,

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1

\Rightarrow \frac{x^2}{5^2}+\frac{y^2}{3^2}=1

\Rightarrow \frac{x^2}{25}+\frac{y^2}{9}=1

Hence the path of the man will be 

\Rightarrow \frac{x^2}{25}+\frac{y^2}{9}=1

 

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