Question

The conic represented by the equation $\sqrt{ax}+\sqrt{by}=1$ is

• A. ellipse
• B. hyperbola
• C. parabola
• D. None of these

Answer 1

Answer: (C)

The given conic is $\sqrt{ax}+\sqrt{by}=1$
Squaring both sides, $ax+by+2\sqrt{abxy}=1$
or $ax+by-1=-2\sqrt{abxy}$
Squaring again, ${\left(ax+by-1\right)}^2=4abxy$
or $a^2{x}^2-2abxy+b^2+y^2-2ax-2by+1=0\dots \left(1\right)$
Comparing the equation $\left(1\right)$ with the equation
$A{x}^2+2Hxy+B{y}^2+2Gx+2Fy+C=0$
$\therefore A=a^2,H=-ab,B=b^2,G=-a,F=-b,C=1$
Then, $\Delta =ABC+2FGH-A{F}^2-B{G}^2-C{H}^2$
$=a^2{b}^2-2a^2{b}^2-a^2{b}^2-a^2{b}^2-a^2{b}^2=-4a^2{b}^2\ne 0$
and $H^2=a^2{b}^2=AB$
So we have $\Delta \ne 0$ and $H^2-AB=0$. Hence the given equation represent a parabola.