Radii of director circles of curves `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` and `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` are 2er and r, respectively. If `e_(1)` and `e_(2)` are the eccentricities of ellipse and hyperbola, respectively, then `4e_(2)^(2)-e_(1)^(2)=`__________.

Options

A. `2e_(h)^(2)-e_(e)^(2)=6`
B. `e_(e)^(2)-4e_(h)^(2)=6`
C. `4e_(h)^(2)-e_(e)^(2)=6`
D. none of these

Explanation

The equation of the director circles of ellipse and hyperbola are
`x^(2)+y^(2)=a^(2)+b^(2)` and `x^(2)+y^(2)=a^(2)-b_(1)^(2)` respectively.
`:.2r=sqrt(a^(2)+b^(2))` and `r=sqrt(a^(2)-b_(1)^(2))`
`=2sqrt(a^(2)-b_(1)^(2))=sqrt(a^(2)+b^(2))`
`implies4(a_(1)^(2)-b_(1)^(2))=a^(2)+b^(2)`
`implies4(1-(b_(1)^(2))/(a^(2)))=(1+(b^(2))/(a^(2)))`
`implies4{1-(e_(h)^(2)-1){=1+(1-e_(e)^(2))`
`implies8-4e_(h)^(2)=2-e_(e)^(2)implies4e_(h)^(2)-e_(e)^(2)=6`