My 400 day Streak problem!

We can write $ax^2+by^2+cxy+dx+ey+f = 0$ as follows:

$\begin{aligned} ax^2+by^2+cxy+dx+ey+f & = 0 \\ x^2+\dfrac {b}{a} y^2+\dfrac {c}{a} xy+ \dfrac {d}{a} x+ \dfrac {e}{a} y+ \dfrac {f}{a} & = 0 \\ \dfrac {b}{a} y^2+\dfrac {c}{a} xy+ \dfrac {d}{a} x+ \dfrac {e}{a} y+ \dfrac {f}{a} & = -x^2 \end{aligned}$

Substituting the values of $(x,y)$ of the five points, we have the following matrix equation.

$\begin{bmatrix} 169 & -130 & -10 & 13 & 1 \\ 0 & 0 & 3 & 0 & 1 \\ 4096 & -832 & -13 & 64 & 1 \\ 64 & 72 & 9 & 8 & 1 \\ 1 & 0 & 0 & -1 & 1 \end{bmatrix} \begin{bmatrix} \frac {b}{a} \\ \frac {c}{a} \\ \frac {d}{a} \\ \frac {e}{a} \\ \frac {f}{a} \end{bmatrix} = \begin{bmatrix} -100 \\ -9 \\ -169 \\ -81 \\ 0 \end{bmatrix}$

$\begin{array} {ll} \begin{bmatrix} \frac {b}{a} \\ \frac {c}{a} \\ \frac {d}{a} \\ \frac {e}{a} \\ \frac {f}{a} \end{bmatrix} & = \begin{bmatrix} 169 & -130 & -10 & 13 & 1 \\ 0 & 0 & 3 & 0 & 1 \\ 4096 & -832 & -13 & 64 & 1 \\ 64 & 72 & 9 & 8 & 1 \\ 1 & 0 & 0 & -1 & 1 \end{bmatrix} ^{-1} \begin{bmatrix} -100 \\ -9 \\ -169 \\ -81 \\ 0 \end{bmatrix} \\ & = \begin{bmatrix} -0.001425 & -0.005809 & 0.000289 & 0.000772 & 0.006173 \\ -0.002849 & -0.035056 & -0.000022 & 0.008488 & 0.029440 \\ -0.017094 & 0.336538 & 0.001068 & -0.018519 & -0.301994 \\ 0.049858 & -0.015425 & -0.002916 & 0.056327 & -0.087844 \\ 0.051282 & -0.009615 & -0.003205 & 0.055556 & 0.905983 \\ \end{bmatrix} \begin{bmatrix} -100 \\ -9 \\ -169 \\ -81 \\ 0 \end{bmatrix} \\ & = \begin{bmatrix} 0.083333333 \\ -0.083333333 \\ 0 \\ -8.916666667 \\ -9 \end{bmatrix} = \begin{bmatrix} \frac{1}{12} \\ -\frac{1}{12} \\ 0 \\ \frac{107}{12} \\ -9 \end{bmatrix} \end{array}$

$\Rightarrow a = 12, b=1, c=-1, d = 0, e = -107, f = -108 \\ \Rightarrow |a|+|b|+ |c| +|d|+|e|+|f|=12+1+1+0+107+108= \boxed{229}$

Note: I have solved the inverse matrix using Excel.