Question

Two straight lines $u=0$ and $v=0$ pass through the origin and angle between them is ${\tan }^{-1}\frac{7}{9}$. If the ratio of the slope of $u=0$ and $v=0$ is $\frac{9}{2}$, then their equations are

• A. $y+3x=0$ and $3y+2x=0$
• B. $2y+3x=0$ and $3y+x=0$
• C. $2y=3x$ and $3y=x$
• D. $y=3x$ and $3y=2x$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $y+3x=0$ and $3y+2x=0$
B $2y+3x=0$ and $3y+x=0$
C $2y=3x$ and $3y=x$
D $y=3x$ and $3y=2x$

Let the slope of $u=0$ be $m$. Then slope of $v=0$ is $\frac{9m}{2}$. Therefore,
$\frac{7}{9}=\left|\frac{m-\frac{9m}{2}}{1+m\times \frac{9m}{2}}\right|=\left|\frac{-7m}{2+9m^2}\right|$
$\Rightarrow 9m^2-9m+2=0$ or $9m^2+9m+2=0$
$m=\frac{2}{3},\frac{1}{3}$ or $m=-\frac{2}{3},-\frac{1}{3}$

Therefore the equations of the lines are
$\left(1\right)3y=x$ and $2y=3x$
$\left(2\right)3y=2x$ and $y=3x$
$\left(1\right)x+3y=0$ and $3x+2y=0$
$\left(1\right)2x+3y=0$ and $3x+y=0$