In this textbook I am studying, they state the following:
The equation of a conic in inhomogeneous coordinates is $$ax^2+bxy+cy^2+dx+ey+f=0$$ ie. a polynomial of degree 2. "Homogenizing" this by the replacements $x:x_1/x_3,y:x_2/x_3,$ gives: $$ax_1^2 + bx_1x_2+cx_2^2+dx_1x_3+ex_2x_3+fx_3^2=0$$ or in matrix form $$\textbf{x}^TC\textbf{x}=0$$ where the conic coefficient matrix C is given by $$\begin{bmatrix} a & b/2 & d/2\\ b/2 & c & e/2\\ d/2 & e/2 & f \end{bmatrix}$$
Is the point in homogenizing this so that we can represent a conic in quadratic form? What is the benefit of such a representation? I am new to homogeneous coordinates so it is a bit confusing to me behind the motivation of it for this particular use case.