If the normals from any point to the parabola x 2 = 4 y cuts the line y = 2 in points whose abscissae are in A.P., then the slopes of the tangents at the 3 conormal points are in

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A point on x²=4y is (2t,t²); the normal there is x+ty=2t+t³. It meets y=2 at x=t³, so the three conormal normals cut y=2 at abscissae t1³, t2³, t3³, given in A.P.: t1³+t3³=2t2³. Conormal ⇒ t1+t2+t3=0. Then t1³+t3³=(t1+t3)³-3t1t3(t1+t3)=-t2³+3t1t2t3=2t2³, so t2²=t1t3. Hence tangent slopes t1,t2,t3 are in G.P.

Mathematics - Parabola Question with Solution | TestHub

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