What conics do the following equations represent? When possible, find their centres, and also their equations referred to the centre.
$6 x^{2} - 5 x y - 6 y^{2} + 14 x + 5 y + 4 = 0 .$

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Solution

In the general equation of a conic i.e. $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
is ellips if $h^{2} - a c < 0$ in the given equation $h^{2} - a c = (5 / 2)^{2} - 4.6 < 0$
Therefore it represents equation of ellips
centre of the ellips can be determined using $\frac{δ S}{δ x} = \frac{δ S}{δ y} = 0$
$12 x - 5 y + 14 = 0$
$12 y + 5 x - 5 = 0$
centre =(-11/3,10/3)

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