Question

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

• A. circles
• B. pair of straight lines
• C. parabolas
• D. rectangular hyperbolas

Answer 1

Answer: (C)

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 $y$ -axis, respectively. If the equation of the curve is
$y=y(x)$, the area $POM$ equals
$\underset{0}{\overset{x}{\int }}ydx$ and the area $PON$
equals $xy-\underset{0}{\overset{x}{\int }}ydx.$
Assuming that $2(POM)=PON$, we therefore have
$2\underset{0}{\overset{x}{\int }}ydx=xy-{\int }_0^{x}ydx$
$\Rightarrow 3\underset{0}{\overset{x}{\int }}ydx=xy$

Differentiating both sides of this gives
$3y=x\frac{dy}{dx}+y$
$\Rightarrow 2y=x\frac{dy}{dx}$
$\Rightarrow \frac{dy}{y}=2\frac{dx}{x}$
$\Rightarrow \log y=2\log x+C$
$\Rightarrow y=C{x}^2$
With $C$ being a constant. This solution represents a parabola. We will get a similar result if we have started instead with $2(PON)=POM$