Question

Find the joint equation of pair of lines through the origin each of which making an angle of ${30}^{\circ }$ with the line $3x+2y-11=0$.

Answer 1

Let m be the slope of one of the lines making an angle of ${30}^{\circ }$ with the 3x+2y-11=0 the angle between the lines having slopes m & m is ${30}^{\circ }$

$tan{30}^{\circ }=|\frac{m-m_1}{1+m{m}_1}|\Rightarrow tan{30}^{\circ }=\frac{1}{\sqrt{3}}$

$\frac{1}{\sqrt{3}}=|\frac{m-(-\frac{3}{2})}{1+m(-\frac{3}{2})}|\Rightarrow \frac{1}{3}=(\frac{2m+3}{2-3m}{)}^2$

$(2-3m{)}^2=3(2m+3{)}^2$

$3m^2+48m+23=0$

$3(\frac{y}{x}{)}^2+48(\frac{y}{x})+23=0$

$3y^2+48xy+23x^2=0$