Question

If the chord through the points whose eccentric angles are α and β on the ellipse `x^2/a^2 + y^2/b^2` = 1 passes through the focus (ae, 0), then the value of tan `α/2 tan  β/2` will be ______.

Options

• `(e + 1)/(e - 1)`

• `(e - 1)/(e + 1)`

• `(e + 1)/(e - 2)`

• `(e - 2)/(e + 1)`

Answer 1

If the chord through the points whose eccentric angles are α and β on the ellipse `x^2/a^2 + y^2/b^2` = 1 passes through the focus (ae, 0), then the value of tan `α/2 tan  β/2` will be `underlinebb((e - 1)/(e + 1))`.

Explanation:

Since the chord passes through focus therefore it is the focal chord of ellipse. Let the coordinates of the endpoints of the focal chord be (acosα, bsinα) and (acosβ. bsinβ). Therefore the equation of the chord is 

`x/acos((α + β)/2) + y/bsin((α + β)/2) = cos((α - β)/2)`

This passes through focus (ae, 0)

∴ `(ae)/a cos ((α + β)/2) = cos((α - β)/2)`

⇒ `(cos((α + β)/2))/(cos((α - β)/2)) = 1/e`

⇒ `(cos((α + β)/2) - cos((α - β)/2))/(cos((α + β)/2) + cos((α - β)/2)) = (1 - e)/(1 + e)`

⇒ `(-2sin  α/2  sin  β/2)/(2cos  α/2  cos  β/2) = (1 - e)/(1 + e)` 

⇒ `-tan  α/2  tan  β/2 = (1 - e)/(1 + e)`

⇒ `tan  α/2  tan  β/2 = (e - 1)/(e + 1)`