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(C) Let $(h, k)$ be any point on the hyperbola $x^2 - y^2 = 8$. Thus,$h^2 - k^2 = 8$. The equations of the asymptotes of the hyperbola $x^2 - y^2 = 8$

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$4$

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(C) Let $(h, k)$ be any point on the hyperbola $x^2 - y^2 = 8$. Thus,$h^2 - k^2 = 8$. The equations of the asymptotes of the hyperbola $x^2 - y^2 = 8$ are $x + y = 0$ and $x - y = 0$. The perpendicular distance $d_1$ from $(h, k)$ to the line $x + y = 0$ is $d_1 = \frac{|h + k|}{\sqrt{1^2 + 1^2}} = \frac{|h + k|}{\sqrt{2}}$. The perpendicular distance $d_2$ from $(h, k)$ to the line $x - y = 0$ is $d_2 = \frac{|h - k|}{\sqrt{1^2 + (-1)^2}} = \frac{|h - k|}{\sqrt{2}}$. The product of the lengths of the perpendiculars is $d_1 d_2 = \frac{|h + k|}{\sqrt{2}} 	imes \frac{|h - k|}{\sqrt{2}} = \frac{|h^2 - k^2|}{2}$. Substituting $h^2 - k^2 = 8$,we get $d_1 d_2 = \frac{8}{2} = 4$.

The product of the lengths of the perpendiculars from any point on the hyperbola $x^2 - y^2 = 8$ to its asymptotes is

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