Two straight lines $u=0$ and $v=0$ passes through the origin and the angle between them is $\tan^{-1}\left(\frac{7}{9}\right)$. If the ratio of slopes of $v=0$ and $u=0$ is $\frac{9}{2}$, then their equations are:
Step-by-Step Solution
Key Concept: Apply the slope constraint equation derived from geometric conditions to obtain a quadratic in one slope variable, then solve for all possible slope pairs.
For quadrilateral $ABCD$ with sides having slopes $m_1$ and $m_2$, use the perpendicularity condition and the constraint $\frac{m_1}{m_2} + \frac{2}{1 + m_1 m_2} = -\frac{7}{9}$ to obtain $9m_2^2 + 9m_2 + 2 = 0$. Solving yields four pairs of slopes: $(m_1, m_2) = \left(\frac{2}{3}, -\frac{1}{3}\right)$, $\left(\frac{1}{3}, -\frac{3}{2}\right)$, $\left(-\frac{2}{3}, -3\right)$, $\left(-\frac{1}{3}, -\frac{3}{2}\right)$, corresponding to lines like $y = 3x$, $3y = 2x$, $3x + y = 0.2x + 3y = 0$, and $x + 3y = 0$.
Correct Answer: 1,2,3,4