Allow us to consider ‘m1’ and ‘m’ be the incline of the two given lines to such an extent that \[\mathbf{m1}\text{ }=\text{ }\mathbf{2m}\]
We realize that in case θ is the point somewhere within l1 and l2 with incline m1 and m2, then, at that point,
\[\mathbf{1}+\mathbf{2m2}\text{ }=\text{ }-\text{ }\mathbf{3m}\]
\[\mathbf{2m2}\text{ }+\mathbf{1}\text{ }+\mathbf{3m}\text{ }=\text{ }\mathbf{0}\]
\[\mathbf{2m}\text{ }\left( \mathbf{m}+\mathbf{1} \right)\text{ }+\text{ }\mathbf{1}\left( \mathbf{m}+\mathbf{1} \right)\text{ }=\text{ }\mathbf{0}\]
\[\left( \mathbf{2m}+\mathbf{1} \right)\text{ }\left( \mathbf{m}+\mathbf{1} \right)=\text{ }\mathbf{0}\]
\[\mathbf{m}\text{ }=\text{ }-\text{ }\mathbf{1}\text{ }\mathbf{or}\text{ }-\text{ }\mathbf{1}/\mathbf{2}\]
In the event that\[\mathbf{m}\text{ }=\text{ }-\text{ }\mathbf{1}\] , the incline of the lines are \[-\text{ }\mathbf{1}\text{ }\mathbf{and}\text{ }-\text{ }\mathbf{2}\]
In the event that\[\mathbf{m}\text{ }=\text{ }-\text{ }\mathbf{1}/\mathbf{2}\] , the incline of the lines are \[-\text{ }\mathbf{1}/\mathbf{2}\text{ }\mathbf{and}\text{ }-\text{ }\mathbf{1}\]
Case 2:
\[\mathbf{2m2}\text{ }\text{ }\mathbf{3m}\text{ }+\text{ }\mathbf{1}\text{ }=\text{ }\mathbf{0}\]
\[\mathbf{2m2}\text{ }\text{ }\mathbf{2m}\text{ }\text{ }\mathbf{m}\text{ }+\text{ }\mathbf{1}\text{ }=\text{ }\mathbf{0}\]
\[\mathbf{2m}\text{ }\left( \mathbf{m}\text{ }\text{ }\mathbf{1} \right)\text{ }\text{ }\mathbf{1}\left( \mathbf{m}\text{ }\text{ }\mathbf{1} \right)\text{ }=\text{ }\mathbf{0}\]
\[\mathbf{m}\text{ }=\text{ }\mathbf{1}\text{ }\mathbf{or}\text{ }\mathbf{1}/\mathbf{2}\]
In the event that\[\mathbf{m}\text{ }=\text{ }\mathbf{1}\] , the incline of the lines are 1 and 2
In the event that\[\mathbf{m}\text{ }=\text{ }\mathbf{1}/\mathbf{2}\] , the incline of the lines are 1/2 and 1
∴ The incline of the lines are \[\left[ -\mathbf{1}\text{ }\mathbf{and}\text{ }-\text{ }\mathbf{2} \right]\text{ }\mathbf{or}\text{ }\left[ -\mathbf{1}/\mathbf{2}\text{ }\mathbf{and}\text{ }-\text{ }\mathbf{1} \right]\text{ }\mathbf{or}\text{ }\left[ \mathbf{1}\text{ }\mathbf{and}\text{ }\mathbf{2} \right]\text{ }\mathbf{or}\text{ }\left[ \mathbf{1}/\mathbf{2}\text{ }\mathbf{and}\text{ }\mathbf{1} \right]\]