Tangents and Areas

Calculus Level pending

A tangent is drawn at a point $P1$ on the curve $y=x^3$ .The tangent intersects the curve again at point $P2$ . Another tangent is drawn at the point $P2$ ,it intersects the curve at $P3$ and so on.Prove that the $abcissae$ of the pts. $P1, P2,...$ are in $GP$ and find the ratio: $\dfrac{Area(\Delta (P1P2P3))}{Area(\Delta(P2P3P4)}$

The answer is 0.0625.

1 solution

Kunal Gupta
Jun 10, 2014

let the pt. P1 be (a,a^3) pt p2 be (b,b^3) now the slope of tangent at p1=3*a^2=(b^3-a^3)/(b-a) solving we get a/b=-0.5 therefore the abcissae are in GP now let P3 be (c,c^3) and P4 be(d,d^3) to calculate the ratio of areas use the determinant formula: finally you'll get ratio:1/16=0.0625

Tangents and Areas

The answer is 0.0625.

1 solution

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