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Identify the conic section $$$- y^{2} - 2 y + \left(2 y - 7\right)^{2} + 7 = 29$$$
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Your Input
Identify and find the properties of the conic section $$$- y^{2} - 2 y + \left(2 y - 7\right)^{2} + 7 = 29$$$.
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, $$$A = 0$$$, $$$B = 0$$$, $$$C = 3$$$, $$$D = 0$$$, $$$E = -30$$$, $$$F = 27$$$.
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
$$$- y^{2} - 2 y + \left(2 y - 7\right)^{2} + 7 = 29$$$A represents a pair of the lines $$$y = 1$$$, $$$y = 9$$$A.
General form: $$$3 y^{2} - 30 y + 27 = 0$$$A.
Factored form: $$$\left(y - 9\right) \left(y - 1\right) = 0$$$A.
Graph: see the graphing calculator.