Question

The partial differential equation of all planes passing through the origin is

Answer 1

Step-by-step explanation:

Let the required equation of the plane be

z=lx+my+nlx+my−z+n=0.....(1)

Now the plane (1) is at constant distance a from the origin

∴a=|n|l2+m2+1−−−−−−−−−√

⟹a=±nl2+m2+1−−−−−−−−−√

Here p=|ax1+by1+cz1+d|a2+b2+c2−−−−−−−−−−√

⟹n=±nl2+m2+1−−−−−−−−−√

∴(1) becomes

lx+my−z±al2+m2+1−−−−−−−−−√=0.....(2)

Differentiating (2) with respect to x and y, we get

l−dzdx=0 and m−dzdy=0

or