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The lengths of the perpendiculars from the focus and the extremities of a focal chord of a parabola on the tangent at the vertex form A. HP B. GP C. AP Answer is GP
consider a parabola y^2=4ax; a>0, with focus (a,0) and tangent to the vertex being x=0now, let a focal chord intersect the parabola at (at^2, 2at) and (ap^2, 2ap)as this chord passes thru focus, we have[2ap-2at]/[ap^2-at^2]=2ap/[ap^2-a]or pt= –1or p^2t^2=1now, the lengths from the tangent at vertex are:at^2, a, ap^2we see that a/at^2=ap^2/aso, the lengths are in GP
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