Let non homogenous curve equation be $ax^2 + 2hxy + by^2 + 2gx + 2fy + c=0 $ then first time partial differentiating with respect to x gives $ ax + hy + g =0$ and second time partial differentiating with respect to y gives $by + hx + f =0$ solving both equations give point of intersection of pair of lines. $$\left(\dfrac{hf-bg}{ab - h^2}\right ) , \left ( \dfrac {gh-af}{ab - h^2}\right)$$

My question is - why partial differentiating makes work easy and what those two equations represent?