A focus of an ellipse is at the origin. The directrix is the line x=4  and the eccentricity is \frac{1}{2} . Then the length of the semi­major axis is

  • Option 1)

    \frac{5}{3}

  • Option 2)

    \frac{8}{3}

  • Option 3)

    \frac{2}{3}

  • Option 4)

    \frac{4}{3}

 

Answers (1)

As we learnt in

Equation of directrices -

x= \pm \frac{a}{e}

- wherein

For the ellipse  

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

 

 and,

 

 

Coordinates of foci -

\pm ae,o

- wherein

For the ellipse  

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

 Distance between focus and directrix:

\frac{a}{e}-ae=4

2a-\frac{a}{2}=4

\frac{3a}{2}=4

a=\frac{8}{3}

 

 


Option 1)

\frac{5}{3}

This option is incorrect.

Option 2)

\frac{8}{3}

This option is correct.

Option 3)

\frac{2}{3}

This option is incorrect.

Option 4)

\frac{4}{3}

This option is incorrect.

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