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

1214558_1427571_ans_7aa3246bbc734233b80a428633712bdd.jpg

Mathematics

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