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(A) The given equation of the pair of lines is $x^2 + 3y^2 + 4xy - 4x - 10y + 3 = 0$. Comparing this with the general form $ax^2 + by^2 + 2hxy + 2gx +

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$\frac{3}{\sqrt{20}}$

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(A) The given equation of the pair of lines is $x^2 + 3y^2 + 4xy - 4x - 10y + 3 = 0$. Comparing this with the general form $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$,we get $a = 1, b = 3, h = 2, g = -2, f = -5, c = 3$. The product of the lengths of the perpendiculars from the origin $(0, 0)$ to the pair of lines represented by $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ is given by the formula $\frac{|c|}{\sqrt{(a-b)^2 + (2h)^2}}$. Substituting the values,we get $\frac{|3|}{\sqrt{(1-3)^2 + (2 	imes 2)^2}} = \frac{3}{\sqrt{(-2)^2 + 4^2}} = \frac{3}{\sqrt{4 + 16}} = \frac{3}{\sqrt{20}}$.

The product of the lengths of the perpendiculars from the origin to the pair of lines $x^2 + 3y^2 + 4xy - 4x - 10y + 3 = 0$ is

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