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Identify the conic section $$$x^{2} \ln\left(4\right) \ln\left(43\right) = \ln\left(415\right)$$$
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Your Input
Identify and find the properties of the conic section $$$x^{2} \ln\left(4\right) \ln\left(43\right) = \ln\left(415\right)$$$.
Solution
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, $$$A = \ln\left(4\right) \ln\left(43\right)$$$, $$$B = 0$$$, $$$C = 0$$$, $$$D = 0$$$, $$$E = 0$$$, $$$F = - \ln\left(415\right)$$$.
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.
Answer
$$$x^{2} \ln\left(4\right) \ln\left(43\right) = \ln\left(415\right)$$$A represents a pair of the lines $$$x = - \frac{\sqrt{\ln\left(256\right)} \sqrt{\ln\left(415\right)}}{2 \ln\left(4\right) \sqrt{\ln\left(43\right)}}$$$, $$$x = \frac{\sqrt{\ln\left(256\right)} \sqrt{\ln\left(415\right)}}{2 \ln\left(4\right) \sqrt{\ln\left(43\right)}}$$$A.
General form: $$$x^{2} \ln\left(4\right) \ln\left(43\right) - \ln\left(415\right) = 0$$$A.
Factored form: $$$\left(2 x \ln\left(4\right) \sqrt{\ln\left(43\right)} - \sqrt{\ln\left(256\right)} \sqrt{\ln\left(415\right)}\right) \left(2 x \ln\left(4\right) \sqrt{\ln\left(43\right)} + \sqrt{\ln\left(256\right)} \sqrt{\ln\left(415\right)}\right) = 0.$$$A
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