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Question: Through any point (x, y) of a curve which passes through the origin, lines are drawn parallel to the...

Through any point (x, y) of a curve which passes through the origin, lines are drawn parallel to the coordinate axes. The curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of

A

Circles

B

Parabolas

C

Hyperbolas

D

Straight lines

Answer

Parabolas

Explanation

Solution

Let P(x, y) be the point on the curve passing through the origin O(0, 0), and let PN and PM be the lines parallel to the x-axes and y-axes, respectively. If the equation of the curve is y = y(x), the area POM equals

dx and the area PON equals xy – dx Assuming that 2 (POM) = PON, we therefore have 2dx = xy – dx

Ž 3dx = xy.

Differentiating both sides of this gives

3y = x + y Ž 2y = x Ž = 2

Ž log |y| = 2 log |x| + C Ž y = Cx2, with C being a constant.

This solutions represents a parabola. We will get a similar result if we had started instead with 2(PON) = POM.