The values of h for which the equation 3x^2 + | vedclass

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

Vedclass · Accepted Answer

$4, 4$

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(A) The general equation of a second-degree curve $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of straight lines if the determinant of the matrix $\begin{bmatrix} a & h & g \ h & b & f \ g & f & c \end{bmatrix}$ is zero.Comparing the given equation $3x^2 + 2hxy - 3y^2 - 40x + 30y - 75 = 0$ with the general form,we get:$a = 3, b = -3, c = -75, h = h, g = -20, f = 15$.The condition for a pair of straight lines is $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$.Substituting the values:$(3)(-3)(-75) + 2(15)(-20)(h) - 3(15)^2 - (-3)(-20)^2 - (-75)(h)^2 = 0$$675 - 600h - 675 + 1200 + 75h^2 = 0$$75h^2 - 600h + 1200 = 0$Dividing by $75$:$h^2 - 8h + 16 = 0$$(h - 4)^2 = 0$Thus,$h = 4, 4$.

The values of $h$ for which the equation $3x^2 + 2hxy - 3y^2 - 40x + 30y - 75 = 0$ represents a pair of straight lines are:

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