Confusing Locus

Calculus Level 3

Consider a family of parabolas y = k x 2 y=kx^2 where k k is Real. Find the locus of all points on the parabola in the f i r s t first quadrant such that the area bounded by the curve, x x -axis and the abscissa of the point is unity.

Rectangular Hyperbola with axis not parallel to coordinate axes Parabola with axis not parallel to coordinate axes Straight Line not parallel to the coordinate axes Circle contained entirely within the first quadrant

Maharnab Mitra by Maharnab Mitra , 25, India

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1 solution

Maharnab Mitra
May 18, 2014

Let the coordinates of the locus be ( X , Y ) (X,Y) so that X = x X=x and Y = k x 2 Y=kx^2 .

Now, the required area is ∫ 0 x k x 2 d x = 1 ⟹ k x 3 3 = 1 ⟹ ( k x 2 ) ( x ) = 3 ⟹ Y X = 3 \int_0^x kx^2\,dx = 1 \implies \frac{kx^3}{3}=1 \implies (kx^2)(x)=3 \implies YX=3 which is a rectangular hyperbola whose axis is not parallel to the coordinat axes.

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