Question

In Exercise 40 through 43, consider the problem of fitting a conic through$\mathit{m}$ given points ${\mathit{P}}_{\mathbf{1}}\mathbf{(}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{y}}_{\mathbf{1}}\mathbf{)}\mathbf{,}\mathbf{.}\mathbf{......}\mathbf{,}{\mathit{P}}_{\mathbf{m}}\mathbf{(}{\mathbf{x}}_{\mathbf{m}}\mathbf{,}{\mathbf{y}}_{\mathbf{m}}\mathbf{)}$in the plane; see Exercise 53 through 62 in section 1.2. Recall that a conic is a curve in${\mathit{ℝ}}^{\mathbf{2}}$ that can be described by an equation of the form $\mathit{f}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}{\mathit{c}}_{\mathbf{1}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}\mathit{x}\mathbf{+}{\mathit{c}}_{\mathbf{3}}\mathit{y}\mathbf{+}{\mathit{c}}_{\mathbf{4}}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{c}}_{\mathbf{5}}\mathit{x}\mathit{y}\mathbf{+}{\mathit{c}}_{\mathbf{6}}{\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{0}$, where at least one of the coefficients is non zero.

42. How many conics can you fit through five distinct points${\mathit{P}}_{\mathbf{1}}\mathbf{(}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{y}}_{\mathbf{1}}\mathbf{)}\mathbf{,}\mathbf{.}\mathbf{......}\mathbf{,}{\mathit{P}}_{\mathbf{5}}\mathbf{(}{\mathbf{x}}_{\mathbf{5}}\mathbf{,}{\mathbf{y}}_{\mathbf{5}}\mathbf{)}$? Describe all possible scenarios, and give an example in each case.

Short answer

There are infinitely many conics that fits through five distinct points.

Step-by-step solution

Given Information

A conic is a curve in${ℝ}^2$ that can be described by an equation of the form $f(x,y)=c_1+c_{2}x+c_{3}y+c_4{x}^2+c_{5}xy+c_6{y}^2=0$, where at least one of the coefficients $c_i$is non-zero.

Step 2:Find the number of conics

To fit a conic through the points$P_1(x_1,y_1),\mathrm{.....},P_5(x_5,y_5)$is equivalent to finding the kernel of an $5x_6$of the matrix so, five points are the solution to the equation.

$f(x,y)=c_1+c_{2}x+c_{3}y+c_4{x}^2+c_{5}xy+c_6{y}^2=0$

Since there are 6 unknowns $c_i\text{'}s$and only 5 points.

Thus, there are infinitely many solutions.

Hence, there are infinitely many conics that fits through five distinct points.