Question
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:
Options
Solution
Sol. [Key Concept: Slope ratio $9:2$ with angle $\tan^{- 1}\frac{79}{}$ gives four sign-combinations of the two line pairs.]
$$\begin{matrix} & \frac{m_{1}}{m_{2}} = \frac{9}{2},\ \frac{m_{1} - m_{2}}{1 + m_{1}m_{2}} = \pm \frac{7}{9} \\ \Rightarrow \ & 9m_{2}^{2} - 9m_{2} + 2 = 0 \\ & 9m_{2}^{2} + 9m_{2} + 2 = 0 \\ & m_{2} = \frac{2}{3},\frac{1}{3}m_{2} = \frac{- 2}{3},\frac{- 1}{3} \\ & m_{2} = \frac{2}{3},m_{1} = 3 \\ \Rightarrow \ & y = 3x,3y = 2x \\ & m_{2} = \frac{1}{3},m_{1} = \frac{3}{2} \\ \Rightarrow \ & 3y = x,2y = 3x \\ & m_{2} = \frac{- 2}{3},m_{1} = - 3 \\ \Rightarrow & 3x + y = 0,2x + 3y = 0 \\ & m_{2} = - \frac{1}{3},m_{1} = \frac{- 3}{2} \\ \Rightarrow \ & x + 3y = 0,3x + 2y = 0 \end{matrix}$$
✔ (a) a valid pair $u = 0,\ v = 0$.
✔ (b) a valid pair.
✔ (c) a valid pair.
✔ (d) a valid pair (all four sign choices work).
Create a free account to view solution
View Solution Free