Find normal lines which pass via origin

[athrun200]

Homework Statement

1. Find all the points P_{0} on the surface z = 2 − xy at which the normal line passes
through the origin.

Homework Equations

The Attempt at a Solution

See the photo.
It seems as long as I can find a,b and c, then the question is done.
However I don't know how to do that

 

[HallsofIvy]

Yes, writing the surface as f(x,y,z)= z+xy= 2 gives the normal at any point (x_0, y_0, z_0) as \nabla f= y_0\vec{i}+ x_0\vec{j}+ \vec{k}. In order to give a line that passes through the origin, that vector must be parallel to the "position" vector of the point: x_0\vec{i}+ y_0\vec{j}+ z_0\vec{k} and so must satisfy y_0= x_0t, x_0= y_0t, and z_0= t for some t. The two equations y_0= x_0t and x_0= y_0t give x_0^2= y_0^2 so that y_0=\pm x_0. With the last, z_0= t we must have t= z_0 and so y_0= x_0t= x_0z_0. Putting those together, y_0= x_0= x_0z_0 would give z_0= 1 so that 1= 2- x_0y_0 x_0y_0= 1 so that x_0= y_0= 1 or x_0= y_0= -1. That is, two such points are (1, 1, 1) and (-1, -1, 1). Or y_0= -x_0= x_0z_0 so we must have z_0= -1 and then x_0y_0= 3 which is not possible since x_0 and y_0 are of different signs.