Trace the parabolas $4 x^{2} - 4 x y + y^{2} - 12 x + 6 y + 9 = 0 .$

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Solution

Given equation of the parabola: $4 x^{2} - 4 x y + y^{2} - 12 x + 6 y + 9 = 0$
Which when simplified gives.
$(- 2 x)^{2} + y^{2} + 3^{2} + 2. (- 2 x) . y + 2. y .3 + 2.3. (- 2 x) = (- 2 x + y + 3)^{2} = 0$
$\Rightarrow 2 x - y - 3 = 0$ which is an equation of a line.
Thus this parabola is traced by 2 coincident lines given by $2 x - y - 3 = 0.$

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