Question

`P_(1)` and `P_(2)` are the lengths of the perpendicular from the foci on the tangent of the ellipse and `P_(3)` and `P_(4)` are perpendiculars from extermities of major axis and P from the centre of the ellipse on the same tangent, then `(P_(1)P_(2)-P^(2))/(P_(3)P_(4)-P^(2))` equals (where e is the eccentricity of the ellipse)
A. e
B. `sqrt(e)`
C. `e^(2)`
D. none of these

Answer 1

Correct Answer - C
Let equation of tangent `y = mx +c`
`P=(|c|)/(sqrt(1+m^(2)))" "P_(1)=(|c+aem|)/(sqrt(1+m^(2)))`
`P_(2)=(|c-aem|)/(sqrt(1+m^(2)))" "P_(3)=(|c+am|)/(sqrt(1+m^(2)))`
`P_(4)=(|c-am|)/(sqrt(1+m^(2)))`
So, `(P_(1)P_(2)-P^(2))/(P_(3)P_(4)-P^(2)) =(c^(2)-a^(2)e^(2)m^(2)-c^(2))/(c^(2)-a^(2)m^(2)-c^(2)) = e^(2)`