How to find PDE of all planes with the following condition

Question

Can someone help with the following problem :

Find the partial differential equation of all planes which are at a constant distance $a$ from the origin.

Thanks in advance for your time.

Answer 1

Let the required equation of the plane be $$z=lx+my+n\\lx+my-z+n=0.....(1)$$Now the plane $(1)$ is at constant distance $a$ from the origin$$\therefore a=\frac{|n|}{\sqrt{l^2+m^2+1}}$$$$\implies a=\frac{\pm n}{\sqrt{l^2+m^2+1}}$$$$\mbox{Here }p=\frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}$$$$\implies n=\frac{\pm n}{\sqrt{l^2+m^2+1}}$$$\therefore (1) $ becomes$$lx+my-z\pm a\sqrt{l^2+m^2+1}=0.....(2)$$Differentiating $(2)$ with respect to $x$ and $y$, we get$$l-\frac{dz}{dx}=0\mbox{ and }m-\frac{dz}{dy}=0$$or$$p=l\mbox{ and }q=m$$$$\therefore(2)\mbox{ reduces to }$$$$px+qy-z\pm a\sqrt{p^2+q^2+1}=0$$$$\implies z=px+qy\pm \sqrt{p^2+q^2+1}\mbox{ is the required differential equation}$$