Question

Trace the following central conics.
$xy=a(x+y).$

Answer 1

$xy=a(x+y)xy-ax-ay=0a=0,b=0,c=0,h=\frac{1}{2},f=-\frac{a}{2},g=-\frac{a}{2}\Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2\Delta =0+2\times \frac{1}{2}\times -\frac{a}{2}\times -\frac{a}{2}-0-0-0=\frac{a^2}{4}\Delta \ne 0h^2=\frac{1}{4},ab=0h^2>ab$

So the equation represents a hyperbola

$\frac{\partial }{\partial x}(xy-ax-ay=0)y-a=\mathrm{0.......}\left(i\right)\frac{\partial }{\partial x}(xy-ax-ay=0)x-a=\mathrm{0.....}\left(ii\right)$

Solvingg $\left(i\right)$ and $\left(ii\right)$ we get the centre of the conic centred at the origin

$C:(a,a)$

Making the conic central by using the centre

$xy=cc=ax+ayc=a\times a+a\times a=2a^{2}xy=2a^2$

which represents a standard hyperbola.