Question

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 1

Answer: (B)

Parabolas

Answer 2

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

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

2 = xy –Ž = 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 solution represents a parabola. We will get a similar result if we had started instead with 2 (PON) = POM.