Identify the conic section $$$12 - 9 x^{2} = - \frac{3 x^{2}}{13} - 16 x + 1$$$

Identify and find the properties of the conic section $$$12 - 9 x^{2} = - \frac{3 x^{2}}{13} - 16 x + 1$$$.

The general equation of a conic section is $$$A x^{2} + B x y + C y^{2} + D x + E y + F = 0$$$.

In our case, $$$A = \frac{114}{13}$$$, $$$B = 0$$$, $$$C = 0$$$, $$$D = -16$$$, $$$E = 0$$$, $$$F = -11$$$.

The discriminant of the conic section is $$$\Delta = 4 A C F - A E^{2} - B^{2} F + B D E - C D^{2} = 0$$$.

Next, $$$B^{2} - 4 A C = 0$$$.

Since $$$\Delta = 0$$$, this is the degenerated conic section.

Since $$$B^{2} - 4 A C = 0$$$, the equation represents two parallel lines.

$$$12 - 9 x^{2} = - \frac{3 x^{2}}{13} - 16 x + 1$$$A represents a pair of the lines $$$x = - \frac{-104 + \sqrt{27118}}{114}$$$, $$$x = \frac{104 + \sqrt{27118}}{114}$$$A.

General form: $$$\frac{114 x^{2}}{13} - 16 x - 11 = 0$$$A.

Factored form: $$$\left(114 x - 104 + \sqrt{27118}\right) \left(114 x - \sqrt{27118} - 104\right) = 0$$$A.

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