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A man running round a racecourse notes that the sum of the distances of two flag-posts from him is always $10 \mathrm{~m}$ and the distance between the flag-posts is $8 \mathrm{~m}$. The area of the path he encloses is
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$15 \pi$ square metres
Given that sum of the distances of two flag-posts from him is always $10 \mathrm{~m}$. So, the race course is in the shape of ellipse. From the given figure, $\mathrm{Mf}_{1}+\mathrm{Mf}_{2}=10$
Let 'a' be the length of semi major axis and 'b' be the length of semi minor axis. $\mathrm{Mf}_{1}+\mathrm{Mf}_{2}=10 \Rightarrow 2 \mathrm{a}=10 \Rightarrow \mathrm{a}=$
Also, $\mathrm{f}_{1} \mathrm{f}_{2}=8$
Let $\mathrm{f}_{1}=(\mathrm{C}, 0)$ and $\mathrm{f}_{2}=(-\mathrm{C}, 0)$.
$\therefore \mathrm{f}_{1} \mathrm{f}_{2}=8 \Rightarrow 2 \mathrm{C}=8 \Rightarrow \mathrm{C}=4$
We knoy $=9=3^{2}$
$\therefore \mathrm{b}=3$
Area of the racecourse $=\pi \mathrm{ab}=\pi \times 5 \times 3=15 \pi$ sq. $\mathrm{m}$
Let 'a' be the length of semi major axis and 'b' be the length of semi minor axis. $\mathrm{Mf}_{1}+\mathrm{Mf}_{2}=10 \Rightarrow 2 \mathrm{a}=10 \Rightarrow \mathrm{a}=$
Also, $\mathrm{f}_{1} \mathrm{f}_{2}=8$
Let $\mathrm{f}_{1}=(\mathrm{C}, 0)$ and $\mathrm{f}_{2}=(-\mathrm{C}, 0)$.
$\therefore \mathrm{f}_{1} \mathrm{f}_{2}=8 \Rightarrow 2 \mathrm{C}=8 \Rightarrow \mathrm{C}=4$
We knoy $=9=3^{2}$
$\therefore \mathrm{b}=3$
Area of the racecourse $=\pi \mathrm{ab}=\pi \times 5 \times 3=15 \pi$ sq. $\mathrm{m}$
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