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Question

A variable line drawn through the point of intersection of the lines x4+y8=1 and x8+y4=1 and meets the coordinate axes at A and B. Locus of mid point of AB is C then
  1. C will be a hyperbola with eccentricity of 2
  2. C will be a hyperbola whose asymptote are x=43,y=43
  3. C always passes through origin
  4. Equation of the director circle of the curve is x2+y2=169


Solution

The correct options are
A C will be a hyperbola with eccentricity of 2
B C will be a hyperbola whose asymptote are x=43,y=43
C C always passes through origin
Intersection point of x4+y8=1 and x8+y4=1 is (83,83)
Let the coordinate of the mid point of AB is (h,k) then 
A(2h,0) and B(0,2k)
A,B,(83,83) are in straight line 
Δ=0Δ=∣ ∣ ∣2h0102k183831∣ ∣ ∣=04hk163h163k=03xy=4(x+y)
Equation of the asymptotes are x=43,y=43
Director circle of the hyperbola is point whose coordinates (43,43)

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