The equation {x^2} - 16xy - 11{y^2} - 12x + 6 | vedclass

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(C) The general equation of a second-degree curve is $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$. Comparing this with the given equation ${x^2} - 16xy -

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Hyperbola

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(C) The general equation of a second-degree curve is $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$. Comparing this with the given equation ${x^2} - 16xy - 11{y^2} - 12x + 6y + 21 = 0$,we get: $a = 1, h = -8, b = -11, g = -6, f = 3, c = 21$. First,we calculate the discriminant $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2$: $\Delta = (1)(-11)(21) + 2(3)(-6)(-8) - (1)(3)^2 - (-11)(-6)^2 - (21)(-8)^2$ $\Delta = -231 + 288 - 9 + 396 - 1344 = -1000 
e 0$. Next,we check the condition $h^2 - ab$: $h^2 - ab = (-8)^2 - (1)(-11) = 64 + 11 = 75$. Since $h^2 - ab > 0$,the equation represents a hyperbola.

The equation ${x^2} - 16xy - 11{y^2} - 12x + 6y + 21 = 0$ represents

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