Let `Sa n dS '` be two foci of the ellipse `(x^2)/(a^3)+(y^2)/(b^2)=1` . If a circle described on `S S^(prime)` as diameter intersects the ellipse at real and distinct points, then the eccentricity`e` of the ellipse satisfies `c=1/(sqrt(2))` (b) `e in (1/(sqrt(2)),1)` `e in (0,1/(sqrt(2)))` (d) none of these

Options

A. `e = 1sqrt2`
B. `e in (1//sqrt2, 1)`
C. `e in (0, 1//sqrt2)`
D. none of these

Answer

B

Explanation

The equation of the circle described on SS' as a diameter is
(x - ae) (x + ae) + (y - 0) (y - 0) = 0
`rArr x^(2) + y^(2) = a^(2)e^(2)`
The abscissae of the points of intersection of the ellipse and this circle are the roots of the equation
`x^(2)/a^(2) + (a^(2)e^(2) - x^(2))/(b^(2)) = 1`
`rArr x^(2) ((1)/(a^(2)) - (1)/(b^(2))) = 1 - (a^(2)e^(2))/(b^(2))`
`rArr x^(2) ((b^(2) - a^(2))/(a^(2)b^(2))) = (b^(2) - a^(2)e^(2))/(b^(2))`
`rArr - (x^(2)a^(2)e^(2))/(a^(2)) = a^(2) - 2a^(2) e^(2)`
`rArr x^(2) = a^(2) ((2e^(2) - 1)/(e^(2))) rArr x = am a/e sqrt(2e^(2) - 1)`
This will give distinct values of x if `2e^(2) - 1 gt 0` i.e. `e gt (1)/(sqrt@).`
Hence, `e in (1//sqrt2, 1)`.