The parametric representation of a point of the ellipse whhose foci are `(3,0) and (-1,0)` and accentricity 2/3 is

Options

A. `(1+3 cos thetasqrt3 sin theta)`
B. `(1+3 cos theta, 5 sin theta)`
C. `(1+3 cos theta, 1+sqrt5sin theta)`
D. `(1+3 cos theta, sqrt5 sin theta)`

Answer

D

Explanation

Given, foci are (3,0) and ( - 1, 0 )
and e = ` ( 2 )/( 3) `.
` therefore 2ae = 4 `
` rArr a = 3 `
Also, ` e ^ 2 = 1 - ( b^ 2 )/( a ^ 2 ) rArr ( 4 )/( 9 ) = 1 - ( b ^ 2 )/( 9) `
` rArr b = sqrt 5 `
since, centre of the ellipse is the mid - point of the line joining the two foci, therefore the coordinates of the centre are (1,0)
` therefore ` Equation of ellipse is
` (( x - 1 )^ 2 )/( 9 ) + (( y - 0 ) ^ 2 )/( 5 ) = 1 `
Hence, the parametric coordinates are ` ( 1 + 3 cos theta , sqrt 5 sin theta ) ` .