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

Dear Student,
Please find below the solution to the asked query:

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

Hope this information will clear your doubts about this topic.

If you have any doubts just ask here on the ask and answer forum and our experts will try to help you out as soon as possible.

  • 64
What are you looking for?