  Question

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

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