Identifying an ellipse

I don't find analytic solution for problem. I use Wolframalpha.

Find ellipse equation in form

$a \cdot x^2+b \cdot y^2+c \cdot x \cdot y+d \cdot x+f \cdot y=1$

For
$y=0$

$a \cdot x^2+d \cdot x=1$

$D=d^2+4a=0$

$a=-\frac{d^2}{4}$

$-\frac{d^2}{4} \cdot x^2+by^2+c\cdot x \cdot y+d \cdot x+f \cdot y=1$

Four equations for points $A(0,6),B(3,9),C(6,8),D(8,3)$

$b \cdot 6^2+f \cdot 6=1$

$-\frac{d^2}{4} \cdot 3^2+b \cdot 9^2+c \cdot 3 \cdot 9+d \cdot 3+f \cdot 9=1$

$-\frac{d^2}{4} \cdot 6^2+b \cdot 8^2+c \cdot 6 \cdot 8+d \cdot 6+f \cdot 8=1$

$-\frac{d^2}{4} \cdot 8^2+b \cdot 3^2+c \cdot 8 \cdot 3+d \cdot 8+f \cdot 3=1$

Solve system

Properties of ellipse 1

Properties of ellipse 2

ellipse1

area enclosed | 102.0215377764767286

ellipse2

area enclosed | 56.64732607727403969317

102.0215377764767286+56.64732607727403969317

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