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 ____________

As we know:

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

Therefore,

a = 6cm and \( a^{2} = 36 \)

b = 4 and \( b^{2} = 16 \)

Therefore,\( b^{2} = a^{2} (1 – e^{2}) \)

=\( e^{2} = 1 – \frac{b^{2}}{a^{2}} = 1 \)

= \( \frac{16}{36} = \frac{5}{9} \)

Therefore,

\( e = \sqrt{\frac{5}{9}} = \sqrt{\frac{5}{3}} \) \( ae = \frac{6 * \sqrt{5}}{3} \)

= \( 2\sqrt{5} \)

Hence, the length of the string and distance between the pins are \( 6 + 2\sqrt{5}, 2\sqrt{5} \)

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