If $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0,\text{ then }\frac{dy}{dx}=?$

  $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

Differentiating w.r.t. $x$ of $y,$ we get

 $2ax+2h\left(y+x\frac{dy}{dx}\right)+2by\frac{dy}{dx}+2g+2f\frac{dy}{dx}=0$

$\therefore \frac{dy}{dx}(2hx+2by+2f)=-(2ax+2hy+2g)$

or $\frac{dy}{dx}=-\frac{(ax+hy+g)}{(hx+by+f)}.$