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Question

A coplanar beam of light emerging from a point source have the equation λxy+2(1+λ)=0, λR; the rays of the beam strike an elliptical surface and get reflected inside the ellipse. The reflected rays form another convergent beam having the equation μxy+2(1μ)=0, μR. Further it is found that the foot of the perpendicular from the point (2,2) upon any tangent to the ellipse lies on the circle x2+y24y5=0.
  1. The eccentricity of the ellipse is equal to 23.
  2. The eccentricity of the ellipse is equal to 13 
  3. The area of the largest triangle that an incident ray and corresponding reflected ray can enclose with major axis of the ellipse is equal to 25.
  4. The area of the largest triangle that an incident ray and corresponding reflected ray can enclose with major axis of the ellipse is equal to 45


Solution

The correct options are
A The eccentricity of the ellipse is equal to 23.
C The area of the largest triangle that an incident ray and corresponding reflected ray can enclose with major axis of the ellipse is equal to 25.
(2y)+λ(x+2)=0 family of lines through (2,2)
And (2,y)+μ(x2)=0 family of lines tharough (2,2)
and x2+y24y5=0 will be auxilary circle & its radius =a=3
Distance between foci =2ac=4
ae=2
e=23

Area of ΔPF1F2=12×base×height=12×4×height
maximum area = maximum height = b

b2=a2(1e2)=9(149)
b=5
Area=25

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