Question

A circle whose diameter is major aixs of ellipe `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1 (agtbgt0)` meets axis at point P. Ifthe orthocentre of `DeltaPF_(1)F_(2) ` lies on ellipse where `F_(1)and F_(2)` are foci of ellipse , then find the eccenricity of the ellipse

Answer 1

Correct Answer - `sqrt((sqrt(5)-1)/(2))`
In the figure, B is orthocentre.
Slope of altitude `S B,m_(1)=(b)/(ae)`
Slope of `SA,m_(2)=(a)/(ae)`
Since `m_(1)xxm_(2)=1`, we have

`(b)/(be)((1)/(-e))=-1`
`rArr 1-e^(2)=e^(4)`
`rArre^(4)+e^(2)-1==0`
`rArre^(2)=(sqrt(5)-1)/(2)`
`rArre=sqrt((sqrt(5)-1)/(2))`