Identify the conic section $$$19 x^{2} = 53$$$

Identify and find the properties of the conic section $$$19 x^{2} = 53$$$.

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, $$$A = 19$$$, $$$B = 0$$$, $$$C = 0$$$, $$$D = 0$$$, $$$E = 0$$$, $$$F = -53$$$.

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.

$$$19 x^{2} = 53$$$A represents a pair of the lines $$$x = - \frac{\sqrt{1007}}{19}$$$, $$$x = \frac{\sqrt{1007}}{19}$$$A.

General form: $$$19 x^{2} - 53 = 0$$$A.

Factored form: $$$\left(19 x - \sqrt{1007}\right) \left(19 x + \sqrt{1007}\right) = 0$$$A.

Graph: see the graphing calculator.