find the locus of the mid points of the chords of the ellipse which are parallel to the line y=2x+c.

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We have:y=2x+cOn comparing with y=mx+c, we get,m=tanθ=2sec2θ=1+tan2θ=1+22=5secθ=5cosθ=15sinθ=tanθ.cosθ=2.15=25Let h,k be the mid-point of the chord.Let the points of intersection of the chord with the ellipse be at a distance r from the midpoint of the chord Intersection points are:Ph+rcosθ,k+rsinθ and Qh-rcosθ,k-rsinθi.e. h+r5,k+2r5 and Qh-r5,k-2r5As these points lie ellipse x2a2+y2b2=1, henceh+r52a2+k+2r52b2=1 ;ih-2r52a2+k-2r52b2=1 ;iii-iih+r52-h-r52a2+k+2r52-k-2r52b2=0We know thata+b2-a-b2=4abHence above equation becomes4.h.r5a2+4.k.2r5b2=04r5ha2+2kb2=0ha2+2kb2=0Hence equation of locus will be:xa2+2yb2=0

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