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

$\begin{array}{1 1}6+\sqrt 5,\sqrt 5\\6-\sqrt 5,\sqrt 5\\6+2\sqrt 5,2\sqrt 5\\6-2\sqrt 5,2\sqrt 5\end{array}$

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A)
Toolbox:
• General equation of an ellipse about the major axis is given by $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1$
• $b^2=a^2(1-e^2)$
Answer : $6+2\sqrt 5,2\sqrt 5$
It is given that the axes are 6 cm and 4 cm.
Hence $a=6\Rightarrow a^2=36$
$b=4\Rightarrow b^2=16$
We know $b^2=a^2(1-e^2)$
(ie) $16=36(1-e^2)$
$\therefore e^2=1-\large\frac{4}{9}=\frac{5}{9}$
$e=\large\frac{\sqrt 5}{3}$
$\therefore ae=\large\frac{6\times \sqrt 5}{3}$
$\Rightarrow 2 \sqrt 5$
Hence the length of the string and distance between the pins are $6+2\sqrt 5,2\sqrt 5$