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(C) Given the equation of the pair of lines: $2x^2 + 4xy - 4y^2 - 6x - 8y + 7 = 0$. To find the length of the intercept on the $Y$-axis,we set $x = 0$

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$\sqrt{11}$

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(C) Given the equation of the pair of lines: $2x^2 + 4xy - 4y^2 - 6x - 8y + 7 = 0$. To find the length of the intercept on the $Y$-axis,we set $x = 0$. Substituting $x = 0$ into the equation,we get: $-4y^2 - 8y + 7 = 0$,which simplifies to $4y^2 + 8y - 7 = 0$. Using the quadratic formula $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$,we have $y = \frac{-8 \pm \sqrt{64 - 4(4)(-7)}}{2(4)} = \frac{-8 \pm \sqrt{64 + 112}}{8} = \frac{-8 \pm \sqrt{176}}{8}$. Since $\sqrt{176} = \sqrt{16 	imes 11} = 4\sqrt{11}$,we get $y = \frac{-8 \pm 4\sqrt{11}}{8} = -1 \pm \frac{\sqrt{11}}{2}$. The length of the intercept is the absolute difference between the two roots $y_1$ and $y_2$: $|y_1 - y_2| = |(-1 + \frac{\sqrt{11}}{2}) - (-1 - \frac{\sqrt{11}}{2})| = |\frac{\sqrt{11}}{2} + \frac{\sqrt{11}}{2}| = \sqrt{11}$.

Find the length of the intercept cut by the pair of lines $2x^2 + 4xy - 4y^2 - 6x - 8y + 7 = 0$ on the $Y$-axis.

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