The ellipse ` E_(1) : x^(2)/9 + y^(2)/4` =1 is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse ` E_(2)` passing through the point (0,4) circumscribes the rectangle R. The eccentricity of the ellispe is

Options

A. `(sqrt(2))/(2)`
B. `(sqrt(3))/(2)`
C. `(1)/(2)`
D. `(3)/(4)`

Answer

C

Explanation

Equilition of an ellipse is `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1 " " [ :' agt b]`
Eccentricity , `e^(2)=1-(b^(2))/(a^(2)) " " [:' agtb]`

Description AS ellipse circumscribes the rectangle then it must pass through all four vertices . Let the equation of an `E_(2)` be
`(x^(2))/(a^(2))+(y^(2))/(b^(2))=1 `, when alt b and b=4 .

also , it passes through (3,2). ,
`implies (9)/(a^(2))+(4)/(b^(2))=1 " " [ :' b=4]`
`implies (9)/(a^(2))+(1)/(4)=1 or a^(2)=12`
`" Eccentricity of " E_(2) e^(2)=1-(a^(2))/(b^(2))=1-(12)/(16)=(1)/(4)[ :' altb]`
`e=(1)/(2)`