Question

Calculation of point of intersection of pair of lines using partial differentiation

Answer 1

It is generally told to us students to mug up this method for find the point of intersection of the pair of straight lines given by: ax2+2hxy+by2+2gx+2fy+c=0 Suppose Φ=ax2+2hxy+by2+2gx+2fy+c, then by partially differentiating Φ with respect to x and again with respect to y gives the two equations: ∂Φ∂x=2(ax+hy+g)∂Φ∂y=2(hx+by+f) Setting them both equal to zero and solving them gives the point of intersection of the two lines represented by the equation Φ=0. So why does this happen? Is there an explanation for it? Or atleast an...