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Conic Sections — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsConic Sections1 Mark
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 ____________.
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Answer

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 $6+2\sqrt{5}$
Solution:
Let equation of the ellipse be $\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1$
According to the question, a = 3 and b = 2
Now, $\text{b}^2=\text{a}^2(1-\text{e}^2)$
$\therefore\ \text{e}^2=2-\frac{\text{b}^2}{\text{a}^2}=1-\frac{4}{9}=\frac{5}{9}$
$\therefore\ \text{e}=\frac{\sqrt{5}}{3}$
From the definition of the ellipse for any point P on the ellipse, we have
SP + S'P = 2a, where S and S' are foci
$\therefore$ Length of the endless string = SP + S'P + SS'
$=2\text{a}+\text{ae}=2(3)+(2)(3)\times\frac{\sqrt{5}}{3}=6+2\sqrt{5}$

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