Question

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

Answer 1

Given: Axes are $6cm$ and $4cm$ long.
So, $a=3$ and $b=2$
Let equation of ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Where $a=3$ and $b=2$
$\Rightarrow e=\sqrt{1-\frac{4}{9}}=\frac{\sqrt{5}}{3}[\because e=\sqrt{1-\frac{b^2}{a^2}}]$
Here pins are foci of the ellipse
$\therefore$ Distance between pins $=2\times 3\times \frac{\sqrt{5}}{3}$
[$\because$ distance between foci $=2ae$ ] $=2\sqrt{5}$
$[\because \text{Length of string}=PS+P{S}^{\prime }+S{S}^{\prime },\text{where}S\text{and}{S}^{\prime }\text{are}foci,\text{and}P\text{is any point on ellipse}]$
$\Rightarrow$ Length of endless string $=2a+2ae$
$=2\times 3+2\sqrt{5}$
$=6+2\sqrt{5}$

Hence, length of string is $6+2\sqrt{5}$ and distance between pins are $2\sqrt{5}$