Question

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

• A. $6,2\sqrt{5}$
• B. $6,\sqrt{5}$
• C. $3,\sqrt{5}$
• D. None of these

Answer 1

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

Let the equation of the ellipse is

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

According to the question $a=3;b=2$

Now $b^2=a^2(1-e)$

$e^2=1-\frac{b^2}{a^2}=1-\frac{4}{9}=\frac{5}{9}$

$e=\frac{\sqrt{5}}{3}$

From the definition of the ellipse

for any point P on the ellipse, we have $SP+S^{\prime }P=2a$

where $S$ and $S^{\prime }$ are foci.

Length of the endless string =$SP+S^{\prime }P$

=$2a+2ae=2\left(3\right)=6$

Distance between the pins=$S{S}^{\prime }$=$2\left(3\right)\frac{\sqrt{5}}{3}=2\sqrt{5}$