A line through the origin has equation y = kx, unless it is the vertical line through the origin, which has equation x = 0.
To find how many times the line y = kx intersects the curve f(x,y) = 0 at the origin, put y = kx and find the multiplicity of the root x=0 in the equation f(x,kx) = 0. To find how many times the line x = 0 intersects the curve f(x,y) = 0 at the origin, put x=0 and find the multiplicity of the root y=0 in the equation f(0,y) = 0.
As an example look at problem (j) in that list. We want to show that there are two lines that intersect the curve more than three times at the origin, and that all other lines through the origin intersect the curve exactly three times there. So put y=kx in the equation of the curve. You get . In this equation, x=0 is a triple root for all values of k except k=0, when it becomes a quadruple root. So apart from the single value k=0, you get a line that meets the curve three times at the origin. So far, we only have one exceptional value. But now look at the vertical line. Put x=0 in the equation of the curve, and it becomes . Here, y=0 is a quadruple root, so that line meets the curve four times at the origin.