Question

Through any point, $(x,y)$ of a curve passing through the origin, lines are drawn parallel to the coordinate axes. The curve divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other. Then the curve may represent a family of:

• A. circles
• B. parabolas
• C. ellipses
• D. hyperbolas

Answer 1

The correct option is B parabolas

There are two cases illustrated in the figure
i) Area $OPB=2($ area of $OAP)$
$\therefore xy-{\int }_0^{x}ydx=2{\int }_0^{x}ydx$
$\Rightarrow xy=3{\int }_0^{x}ydx$
Differentiation w.r.t.x.
$y+x\frac{dy}{dx}=3y\Rightarrow \frac{dy}{dx}=2\frac{dx}{x}$
This an integrating yields $y={cx}^2$ similarly (ii) yields $x={cy}^2$