The joint equation of a pair of lines passing | vedclass

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Question

(A) Let the two lines be $OA$ and $OB$ passing through the origin $O(0,0)$.Since each line makes an angle of $30^{\circ}$ with the $Y$-axis,the angles

Vedclass · Accepted Answer

$3x^2-y^2=0$

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(A) Let the two lines be $OA$ and $OB$ passing through the origin $O(0,0)$.Since each line makes an angle of $30^{\circ}$ with the $Y$-axis,the angles they make with the positive $X$-axis are $90^{\circ}-30^{\circ}=60^{\circ}$ and $90^{\circ}+30^{\circ}=120^{\circ}$.The slopes of these lines are $m_1 = 	an(60^{\circ}) = \sqrt{3}$ and $m_2 = 	an(120^{\circ}) = -\sqrt{3}$.The equations of the lines are $y = \sqrt{3}x$ and $y = -\sqrt{3}x$,which can be written as $\sqrt{3}x - y = 0$ and $\sqrt{3}x + y = 0$.The joint equation of the pair of lines is given by $(\sqrt{3}x - y)(\sqrt{3}x + y) = 0$.Using the identity $(a-b)(a+b) = a^2 - b^2$,we get $(\sqrt{3}x)^2 - y^2 = 0$,which simplifies to $3x^2 - y^2 = 0$.

The joint equation of a pair of lines passing through the origin,each of which makes an angle of $30^{\circ}$ with the $Y$-axis,is

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