Question
The sum of the squares of the perpendicular on any tangent to the ellipse 
+ 
= 1 from two points on the minor axis each distance 
from the centre is -
- a2
- b2
- 2a2
- 2b2
Hint:
find the equation of tangent and find the distances of the points from the tangent by using the distance equation.
The correct answer is: 2a2
2a2
Equation of tangent : bx cos Ѳ+ ay sin Ѳ= ab
Points on the minor axis are: (0,√a2-b2) and (0,-√a2-b2)
Distance of perpendicular from these 2 points on the tangent are :
a(√a2-b2sinѲ – b)/ √b2cos2Ѳ+a2sin2Ѳ and a(-√a2-b2sinѲ – b)/ √b2cos2Ѳ+a2sin2Ѳ
sum of squares of the lengths = 2a2(b2cos2Ѳ+a2sin2Ѳ)/ (b2cos2Ѳ+a2sin2Ѳ)
= 2a2
the y axis is the minor axis in this case hence the points are (0,√a2-b2) and (0,-√a2-b2)
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