Question:
If a tangent at a point P1 other than origin on the curve y=x^3 meets the curve again at P2 and the tangent at P2 meets the curve again at P3 and so on,then the abscissae of P1,P2,...,Pn form a option: 1)G.P. 2)A.P. 3) H.P. 4) none of these
Answer:
Let a point P1 on y = x3 be (h, h3 )
Tangent at P1 is
y - h3 = 3h2 (x - h)
It meets y = x3 at P2
=> x3 - h3 = 3h2 (x - h)
=> (x - h)*(x2 + xh + h2 ) = 3h2 (x - h)
=> x2 + xh + h2 = 3h2
=> x2 + xh + h2 - 3h2 = 0
=> x2 + xh - 2h2 = 0
=> (x - h)*(x + 2h) = 0
=> x = -2h for P2 as x = h for P1
So, P2 is (-2h, -8h3 )
Now,
Tangent at P2 is
y + 8h3 = 3(2h)2 (x + 2h)
It meets y = x3 at P3
=> x3 + 8h3 = 12h2 (x + 2h)
=> x2 - 2xh - 8h2 = 0
=> (x - 4h)*(x + 2h) = 0
=> x = 4h for P3
So, P3 is (4h, 64h3 )
Continue like this, we get x = -8h for P4 , etc.
Hence, the abscissae of P1 , P2 , P3 , ........ are h, -2h, 4h, -8h, ...........which is in G.P.
So, abscissae of P1 , P2 , P3 , ........ form a G.P.
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