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Normals are drawn at points `A, B, and C` on the parabola `y^2 = 4x` which intersect at P. The locus of the point P if the slope of the line joining the feet of two of them is 2, is |
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Answer» The equation of normal at `(r^(2),2t)` is `y=-tx+2t+t^(3)` If it passes through P(h,k), then `t^(3)+t(2-h)-k=0` (1) This equation has roots `t_(1),t_(2)andt_(3)` which are parameters of three feet of normal has three feet of normals on the parabola. From equation (1), we have `t_(1)+t_(2)+t_(3)=0` (2) Given that slope of the line joining the feet of two the normals (say `A(t_(1))andB(t_(2))` is 2. `:." "(2)/(t_(1)+t_(2))=2` `rArr" "t_(1)+t_(2)=1` So, from (2), we have `t_(3)=-1`, which is one of the roots of equation (1). So, from (1) we have -1+h-2-k=0 or x-y-3=0, which is required locus. |
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