Trace the following central conics.
$x^{2} - x y + 2 y^{2} - 2 a x - 6 a y + 7 a^{2} = 0 .$

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Solution

Trace the conic
$x^{2} - x y + 2 y^{2} - 2 a x - 6 a y + 7 a^{2} = 0 (2 x - y - 2 a) = \pm \sqrt{- 7 y^{2} + 28 a y - 28 {a a}^{2} + 4 a^{2}} (2 x - y - 2 a)^{2} = - 7 (y^{2} + 4 a^{2} - 4 a y) + 4 a^{2} (2 x - y - 2 a)^{2} + 7 (y - 2 a)^{2} = 4 a^{2}$

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