A Proof
Our two pencils lie on two points, say (0,0) and (0,c). Then the
lines in one pencil will be y=mx+b, and the lines in the other
will be y=(pm+q)/(rm+s)x+c (since the slopes must be related by a
fractional linear transformation).
Then m=y/x. So y=(py+qx)/(ry+sx)x+c;
(y-c)/x=(py+qx)/(ry+sx).
A crossmultiplication yields a quadratic in x and y on both sides.
When we bring everything to one side, our equation states that a
quadratic function of x and y is equal to 0, and this is one way of
defining a conic in analytic geometry.