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Question

If the normals from any point to the parabola y2=4x cuts the line x=2 in the points whose ordinates are in A.P, then What will be the relationship between slopes of tangents at the co-normal points?

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Solution

We know that the equation of normal for y2=4x is y=xt+2t+t3 ....(1)
Since it intersects at x=2, we get y=t3
Let the three ordinates be t31,t22,t33 are in A.P.
2t32=t31+t33
=(t1+t3)33t1t3(t1+t3) ....(2)
Now, t1+t2+t3=0
t1+t3=t2
Hence, Eq.(2) reduces to
2t32=(t2)33t1t3(t2)
2t32=t32+3t1t2t3
3t32=3t1t2t3
t22=t1t3
So, t1,t2,t3 are in G.P.
Hence, slopes of tangents 1t1, 1t2, 1t3 are in G.P.

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