Question

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

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 $x$- and $y$-axes, respectively. If the equation of the curve is $y=y\left(x\right)$,
the area $POM$ equals ${\int }_0^{x}ydx$ and the area $PON$ equals $xy-{\int }_0^{x}ydx$.
Assuming that $2\left(POM\right)=PON$, we therefore have $2{\int }_0^{x}ydx=xy-{\int }_0^{x}ydx\Rightarrow 3{\int }_0^{x}ydx=xy$.
Differentiating both sides of this gives
$3y=x\frac{dy}{dx}+y\Rightarrow 2y=x\frac{dy}{dx}\Rightarrow \frac{dy}{y}=2\frac{dx}{x}$
$\Rightarrow \log \left|y\right|=2\log \left|x\right|+C\Rightarrow y=C{x}^2$ with $C$ being a constant.
This solution represents a parabola.