The equation 3x^2 + 7xy + 2y^2 + 5x + 5y + 2 | vedclass

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(C) The general second-degree equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of straight lines if the determinant $\Delta = abc +

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$A$ hyperbola

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(C) The general second-degree equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of straight lines if the determinant $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0$.Comparing the given equation $3x^2 + 7xy + 2y^2 + 5x + 5y + 2 = 0$ with the general form:$a = 3, b = 2, c = 2, h = 7/2, g = 5/2, f = 5/2$.Now,calculate the determinant $\Delta$:$\Delta = (3)(2)(2) + 2(5/2)(5/2)(7/2) - 3(5/2)^2 - 2(5/2)^2 - 2(7/2)^2$$\Delta = 12 + 2(125/8) - 3(25/4) - 2(25/4) - 2(49/4)$$\Delta = 12 + 125/4 - 75/4 - 50/4 - 98/4$$\Delta = 12 + (125 - 75 - 50 - 98)/4 = 12 - 98/4 = 12 - 24.5 = -12.5 
eq 0$.Since $\Delta 
eq 0$,the equation does not represent a pair of straight lines. Checking the discriminant $h^2 - ab = (7/2)^2 - (3)(2) = 49/4 - 6 = 12.25 - 6 = 6.25 > 0$. Since $h^2 - ab > 0$,it represents a hyperbola.

The equation $3x^2 + 7xy + 2y^2 + 5x + 5y + 2 = 0$ represents:

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