Through any point $(x, y)$ of a curve which passes through the origin, lines are drawn parallel to the co-ordinate 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:
Step-by-Step Solution
Key Concept: Set up area relationships using integration: the area under the curve from origin to point (x,y) is ∫₀ˣ y dx, and the rectangular area minus this gives xy - ∫₀ˣ y dx. The condition that one area is twice the other translates to a differential equation that yields dy/dx = 2y/x.
Given that $P(x,y)$ lies on a curve through origin with $PN$ and $PM$ parallel to axes, the area $POM = \int_0^x y\,dx$ and $PON = xy - \int_0^x y\,dx$. If $2(POM) = PON$, then $3\int_0^x y\,dx = xy$. Differentiating gives $3y = x\frac{dy}{dx} + y$, so $2y = x\frac{dy}{dx}$. Solving this separable equation: $\frac{dy}{y} = 2\frac{dx}{x}$ yields $\log y = 2\log x + C$, giving $y = Cx^2$, which is a parabola.
Correct Answer: 3