Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it ?

Options

A. `(4sqrt(2),2sqrt(3))`
B. `(4sqrt(3),2sqrt(2))`
C. `(4sqrt(2),2sqrt(2))`
D. `(4sqrt(3),2sqrt(3))`

Answer

B

Explanation

Let the equilibrium of ellipse be `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
then , according the problem , we have
`(2b^(2))/(a)=8 and 2ae=2b`
[Length of laturectum `=(2b^(2))/(a) and `length of minor axis of minor axis = 2b]
`implies b((b)/(a) )=4 and (b)/(a)=e`
`implies b(e)=4`
`implies b=4.(1)/(e)`
Also we know that `b^(2)=a^(2)(1-e^(2))`
`implies (b^(2))/(a^(2))=1-e^(2)implies e^(2)=1-e^(2)[:' (b) /(a)=e]`
`implies 2e^(2)=1`
`e=(1)/(sqrt(2))`... . (ii)
from Eqs. (i) and(ii) we get
`b=4sqrt(2)`
`"Now, " a^(2)=(b^(2))/(1-e^(2))=(32)/( 1-(1)/(2))=64`
` therefore` Equation of ellipse be `(x^(2))/(64)+(y^(2))/(32)=1`
Now , check all the options.
only `(4sqrt(3),2sqrt(2)),`satisfy the above equation.