Question

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

• A. $6+2\sqrt{5}$
• B. $6-2\sqrt{5}$
• C. $3+\sqrt{5}$
• D. $3-\sqrt{5}$

Answer 1

The correct option is A $6+2\sqrt{5}$

We know $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
$\therefore a=6㎝,a^2=36$
$b=4,\therefore b^2=16$
$\therefore b^2=a^2(1-e^2)$
$\Rightarrow e^2=1-\frac{b^2}{a^2}=1-\frac{16}{36}=\frac{5}{9}$
$\therefore e=\sqrt{\frac{5}{9}}=\frac{\sqrt{5}}{3}$
$\therefore ae=\frac{6\times \sqrt{5}}{3}$
$=2\sqrt{5}$
$\therefore$ Length of the string and distance between the pins are $6+2\sqrt{5},2\sqrt{5}$