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(D) The vertex $O$ is the intersection of $4x + 5y = 0$ and $7x + 2y = 0,$ which is the origin $(0, 0).$Since the diagonal $11x + 7y = 9$ does not pas

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None of these

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(D) The vertex $O$ is the intersection of $4x + 5y = 0$ and $7x + 2y = 0,$ which is the origin $(0, 0).$Since the diagonal $11x + 7y = 9$ does not pass through the origin $(0, 0),$ it cannot be the diagonal $OB.$ Thus,$11x + 7y = 9$ is the equation of the diagonal $AC.$Let the sides be $OA: 4x + 5y = 0$ and $OC: 7x + 2y = 0.$Solving $AC$ with $OA$: $4x + 5y = 0$ and $11x + 7y = 9.$ Multiplying the first by $7$ and the second by $5$: $28x + 35y = 0$ and $55x + 35y = 45.$ Subtracting gives $27x = 45 \implies x = \frac{5}{3}.$ Then $y = -\frac{4}{3}.$ So $A = \left(\frac{5}{3}, -\frac{4}{3}\right).$Solving $AC$ with $OC$: $7x + 2y = 0$ and $11x + 7y = 9.$ Multiplying the first by $7$ and the second by $2$: $49x + 14y = 0$ and $22x + 14y = 18.$ Subtracting gives $27x = -18 \implies x = -\frac{2}{3}.$ Then $y = \frac{7}{3}.$ So $C = \left(-\frac{2}{3}, \frac{7}{3}\right).$The midpoint $M$ of diagonal $AC$ is $\left(\frac{5/3 - 2/3}{2}, \frac{-4/3 + 7/3}{2}\right) = \left(\frac{1/3}{2}, \frac{3/3}{2}\right) = \left(\frac{1}{6}, \frac{1}{2}\right).$The other diagonal $OB$ passes through the origin $(0, 0)$ and the midpoint $M\left(\frac{1}{6}, \frac{1}{2}\right).$The slope of $OB$ is $m = \frac{1/2 - 0}{1/6 - 0} = \frac{1/2}{1/6} = 3.$The equation of $OB$ is $y = 3x,$ or $3x - y = 0.$ Since this is not among the options,the correct answer is $D$ (None of these).

Two consecutive sides of a parallelogram are $4x + 5y = 0$ and $7x + 2y = 0.$ If the equation to one diagonal is $11x + 7y = 9,$ then the equation of the other diagonal is

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