Chapter 3.3, Problem 42E

In Exercises 40 through 43, consider the problem of fitting a conic through m given points $P_1\left(x_1,y_1\right),\mathrm{...},P_m\left(x_m,y_m\right)$in the plane; see Exercises 53 through 62 in Section 1.2. Recall that a conic is a curve in ${ℝ}^2$that can be described by an equation of the form $f\left(x,y\right)=c_1+c_{2}x+c_{3}y+c_4{x}^2+c_{5}xy+c_6{y}^2=0$, where at least one of the coefficients $c_i$is nonzero.

42. How many conics can you fit through five distinct points $P_1\left(x_1,y_1\right),\mathrm{...},P_5\left(x_5,y_5\right)$ ? Describe all possible scenarios, and give an example in each case.