Question

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

Solution

## 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 $$30^{\circ}$$$$tan30^{\circ}=|\frac{m-m_{1}}{1+mm_{1}}|\Rightarrow tan 30^{\circ}=\frac{1}{\sqrt{3}}$$$$\frac{1}{\sqrt{3}}=|\frac{m-(-\frac{3}{2})}{1+m(-\frac{3}{2})}|\Rightarrow \frac{1}{3}=(\frac{2m+3}{2-3m})^{2}$$$$(2-3m)^{2}=3(2m+3)^{2}$$$$3m^{2}+48m+23=0$$$$3(\frac{y}{x})^{2}+48(\frac{y}{x})+23=0$$$$3y^{2}+48xy+23x^{2}=0$$Mathematics

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