The curve represented by the equation $\sqrt{\mathrm{px}}+\sqrt{\mathrm{qy}}=1$,where $\mathrm{p},\mathrm{q}\in \mathrm{R},\mathrm{p},\mathrm{q}>0$ is

We have $\sqrt{\mathrm{px}}+\sqrt{\mathrm{qy}}=1$

$\begin{array}{l}\mathrm{or} (\sqrt{\mathrm{px}}+\sqrt{\mathrm{qy}}{)}^2=1\\ \mathrm{or} \mathrm{px}+\mathrm{qy}+2\sqrt{\left(\mathrm{pq}\right)\left(\mathrm{xy}\right)}=1\\ \mathrm{or} (\mathrm{px}+\mathrm{qy}-1{)}^2=4(\mathrm{pq}\left)\right(\mathrm{xy})\\ \mathrm{or} {\mathrm{p}}^2{\mathrm{x}}^2-2\left(\mathrm{pq}\right)\left(\mathrm{xy}\right)+{\mathrm{q}}^2{\mathrm{y}}^2-2\mathrm{px}-2\mathrm{qy}+1=0\end{array}$

On comparing this equation with the equation

${\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0$

we get,

$\begin{array}{l}\mathrm{a}={\mathrm{p}}^2,\mathrm{b}={\mathrm{q}}^2,\mathrm{c}=1,\mathrm{g}=-\mathrm{p},\mathrm{f}=-\mathrm{q},\text{ and }\mathrm{h}=-\mathrm{pq} \\ \\ \therefore \mathrm{\Delta }=\mathrm{abc}+2\mathrm{fgh}-{\mathrm{af}}^2-{\mathrm{bg}}^2-{\mathrm{ch}}^2\\ ={\mathrm{p}}^2{\mathrm{q}}^2-2{\mathrm{p}}^2{\mathrm{q}}^2-{\mathrm{p}}^2{\mathrm{q}}^2-{\mathrm{p}}^2{\mathrm{q}}^2-{\mathrm{p}}^2{\mathrm{q}}^2=-4{\mathrm{p}}^2{\mathrm{q}}^2\ne 0\\ \text{ and }{\mathrm{h}}^2-\mathrm{ab}={\mathrm{p}}^2{\mathrm{q}}^2-{\mathrm{p}}^2{\mathrm{q}}^2=0\end{array}$

Thus, we have $\mathrm{\Delta }\ne 0\text{ and }{\mathrm{h}}^2=\mathrm{ab}$

Hence, the given curve is a parabola.