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Question

(A) The center of the rectangle is the midpoint of the diagonal joining $(0, 0)$ and $(8, 6)$,which is $(\frac{0+8}{2}, \frac{0+6}{2}) = (4, 3)$.The l

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$3x - 4y \pm 25 = 0$

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(A) The center of the rectangle is the midpoint of the diagonal joining $(0, 0)$ and $(8, 6)$,which is $(\frac{0+8}{2}, \frac{0+6}{2}) = (4, 3)$.The length of the diagonal is $\sqrt{(8-0)^2 + (6-0)^2} = \sqrt{64 + 36} = 10$.The radius of the circumcircle is $R = \frac{10}{2} = 5$.The equation of the circumcircle is $(x-4)^2 + (y-3)^2 = 5^2$,which simplifies to $x^2 + y^2 - 8x - 6y = 0$.The slope of the diagonal is $m = \frac{6-0}{8-0} = \frac{6}{8} = \frac{3}{4}$.The tangents parallel to the diagonal have the form $y = \frac{3}{4}x + c$,or $3x - 4y + 4c = 0$.The distance from the center $(4, 3)$ to the tangent is equal to the radius $R = 5$.Using the distance formula: $\frac{|3(4) - 4(3) + 4c|}{\sqrt{3^2 + (-4)^2}} = 5$.$\frac{|12 - 12 + 4c|}{5} = 5 \implies |4c| = 25 \implies 4c = \pm 25$.Substituting $4c$ back into the equation $3x - 4y + 4c = 0$,we get $3x - 4y \pm 25 = 0$.

The diagonals of a rectangle have endpoints at $(0, 0)$ and $(8, 6)$. The equations of the tangents to the circumcircle of the rectangle,which are parallel to these diagonals,are:

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