Question

An ellipse is described by using an endless string which passed over two points. If the axes are 6 cm and 4 cm, the length of the string and distance between the points are ___________.

Answer 1

Let equation of ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Length of major axis i.e 2a = 6
i.e a = 3
and Length of minor axis is 4 = 2b
i.e b = 2
$\therefore$ eccentricity $e=1-\frac{b^2}{a^2}$
i.e $e=1-\frac{4}{9}=\frac{5}{9}$
Let S and S' represents the foci of ellipse and P be any point an ellipse
$\therefore$ SP + S'P = 2a
$\therefore$ Length of the endless string SP + S'P + SS'
= 2a + 2ae
$=2\times 3+2\times 3\times \frac{\sqrt{5}}{3}$
i.e Length of the endless string = 6 + $2\sqrt{5}$ and distance between the points is SS' = $2\sqrt{5}$