Section 1
Introduction
$$z=a x^{2}+b y^{2}$$
$$(x-a)^{2}+(y-b)^{2}=z^{2} \cot ^{2} \alpha$$
$$z=a x+b y+a^{2}+b^{2}$$
$$z=a x y+b$$
$$z=\frac{1}{2}(\sqrt{x+a}+\sqrt{y-a}+b)$$
$$u=a(x+y)+b(x-y)+a b z+c$$
$$z=x y+y \sqrt{x^{2}-a^{2}-b^{2}}$$
$$z=a x e^{y}+\frac{1}{2} a^{2} e^{2 y}+b$$
$$z=a x+b y+\left(\frac{a}{b}\right)-b$$
$$z=a \log \left[\frac{b(y-1)}{(1-x)}\right]$$
Form the partial differential equation of all spheres of radius $a$ with their centres on the $x-y$ plane.
Form the partial differential equation of all planes through the origin.