A man running round a racecourse notes that the sum of the...

Given that sum of the distances of two flag-posts from him is always 10 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. $$\begin{aligned} & \therefore \mathrm{Mf}_1+\mathrm{Mf}_2=10 \Rightarrow 2 \mathrm{a}=10 \Rightarrow \mathrm{a}=5 \\ & \text{Also, } \mathrm{f}_1 \mathrm{f}_2=8 \\ & \text{Let } \mathrm{f}_1=(\mathrm{C}, 0) \text{ and } \mathrm{f}_2=(-\mathrm{C}, 0)\end{aligned}$$\therefore \mathrm{f}_1 \mathrm{f}_2=8 \Rightarrow 2 \mathrm{C}=8 \Rightarrow \mathrm{C}=4$We know,$\mathrm{a}^2=\mathrm{b}^2+\mathrm{c}^2 \Rightarrow 5^2=\mathrm{b}^2+4^2$$\begin{aligned}& \Rightarrow \mathrm{b}^2=25-16=9=3^2 \\ & \therefore \mathrm{ b}=3 .\end{aligned}$$$