First slide

Cartesian plane

Question

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

Moderate

A

$3 x + 7 y = 0$
B

$11 x - 7 y = 0$
C

$x - y = 0$
D

$x + y = 0$

Solution

$Given equations are 4 x + 5 y = 0 \dots \dots (i)$
$\begin{aligned} 7 x + 2 y = 0 \dots \dots (ii) \\ 11 x + 7 y = 9 \dots \dots (iii) \\ Solving (i) and (iii) \end{aligned}$
$\begin{aligned} A \equiv (\frac{5}{3}, - \frac{4}{3}) \\ Solving (ii) and (iii) \\ C \equiv (- \frac{2}{3}, \frac{7}{3}) \end{aligned}$
$\begin{array}{l} M = midpoint of A C = (\frac{\frac{5}{3} - \frac{2}{3}}{2}, \frac{\frac{- 4}{3} + \frac{7}{3}}{2}) \\ = (\frac{1}{2}, \frac{1}{2}) \\ Equation of diagonal O B is \end{array}$
$\begin{array}{l} y - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} (x - x_{1}) \\ \Rightarrow y - 0 = \frac{\frac{1}{2} - 0}{\frac{1}{2} - 0} (x - 0) \\ \Rightarrow x - y = 0 \end{array}$
Therefore, the correct answer is (3).

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