Mathematics - Straight Lines Question with Solution | TestHub

The equations of the lines are $$\begin{array}{l}{\mathrm{p}}_1\mathrm{x}+{\mathrm{q}}_1\mathrm{y}-1=0\\ {\mathrm{p}}_2\mathrm{x}+{\mathrm{q}}_2\mathrm{y}-1=0\end{array}$$ and ${\mathrm{p}}_3\mathrm{x}+{\mathrm{q}}_3\mathrm{y}-1=0$ As they are concurrent, $$\left|\begin{array}{lll}{\mathrm{p}}_1& {\mathrm{q}}_1& -1\\ {\mathrm{p}}_2& {\mathrm{q}}_2& -1\\ {\mathrm{p}}_3& {\mathrm{q}}_3& -1\end{array}\right|=0\Rightarrow \left|\begin{array}{lll}{\mathrm{p}}_1& {\mathrm{q}}_1& 1\\ {\mathrm{p}}_2& {\mathrm{q}}_2& 1\\ {\mathrm{p}}_3& {\mathrm{q}}_3& 1\end{array}\right|=0$$ This is also the condition for the points $\left({\mathrm{p}}_1,{\mathrm{q}}_1\right),\left({\mathrm{p}}_2,{\mathrm{q}}_2\right)$ and $\left({\mathrm{p}}_3,{\mathrm{q}}_3\right)$ to be collinear.