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Analytical Geometry

Blazing goIITian

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4 Apr 2009 20:27:04 IST

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Let y=f(x) be a curve. Lines are drawn through any point (x,y) on the curve parallel to the axes of

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Let y=f(x) be a curve. Lines are drawn through any point (x,y) on the curve parallel to the axes of refrence the rectangle formed by the axes of refrence and the lines drawn in two parts, the ratio of whose areas in 2:1 then prove that the curve must be a parabola.

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Prakhar Banga

Blazing goIITian

Joined: 20 Dec 2008

Posts: 599

4 Apr 2009 20:47:14 IST

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Here it doesn't matter which part is 2 and which is 1 because everything else is symmetric. Let us say that the area bounded by the curve and the x axis is 2/3 times the area of the rectangle.

Let (h, k) be a point on the curve.

The area of the rectangle formed will be hk.

The area under the curve above the x axis will be integral 0 to h of f(x) dx

Differentiating both sides w.r.t h

f(h) = 2k/3 + 2hk'/3

k=2k/3 + 2hk'/3

k = 2hk'

k' / k = 1/(2h)

dy / (dx y) = 1/2x

dy / y = dx/2x

Integrating both sides, ln y = ln x / 2 + C

Or y^2 = Cx, which is a parabola

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