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Question

The eccentricity of an ellipse is $$\dfrac {\sqrt {3}}{2}$$ its length of latus reetum is


A
12 (length of major axis)
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B
13 (length of major axis)
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C
14 (length of major axis)
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D
23 (length of major axis)
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Solution

The correct option is B $$\dfrac {1}{4}$$ (length of major axis)
Eccentricity of ellipse is $$e=\frac { c }{ a }$$
$$\therefore \frac { \sqrt { 3 }  }{ 2 } =\frac { c }{ a }$$        (Given)

$$\therefore c=\sqrt { 3 } $$ and $$a=2$$

By property of ellipse, $${ c }^{ 2 }={ a }^{ 2 }-{ b }^{ 2 }$$
$$\therefore { \left( \sqrt { 3 }  \right)  }^{ 2 }={ \left( 2 \right)  }^{ 2 }-{ b }^{ 2 }$$
$$\therefore 3=4-{ b }^{ 2 }$$
$$\therefore { b }^{ 2 }=1$$
$$\therefore { b }=1$$

Length of latus rectum is given by,
$$L=\frac { 2{ b }^{ 2 } }{ a } $$
$$\therefore L=\frac { 2{ \left( 1 \right)  }^{ 2 } }{ 2 }$$
$$\therefore L=1$$

Length of major axis is $$2a=2\times 2=4$$

Thus, we can conclude that,
$$L=\frac { 1 }{ 4 } \left( Length\ of\ major\ axis \right)$$

Mathematics

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