Question

21) Find the equation of the line perpendicular to the line whose equation is 6x - 7y + 8 = 0 and that passes through the point of intersection of the two lines whose equations are 2x - 3y - 4 = 0 and 3x + 4y - 5 = 0.

Answer 1

Dear student
$$\begin{gathered} \mathrm{The}\mathrm{eq}\mathrm{of}\mathrm{any}\mathrm{line}\mathrm{through}\mathrm{the}\mathrm{intersection}\mathrm{of}\mathrm{the}\mathrm{lines}2\mathrm{x}-3\mathrm{y}-4=0 \\ \mathrm{and}3\mathrm{x}+4\mathrm{y}-5=0\mathrm{is} \\ \left(2\mathrm{x}-3\mathrm{y}-4\right)+\mathrm{\lambda }\left(3\mathrm{x}+4\mathrm{y}-5\right)=0 \\ \mathrm{x}\left(2+3\mathrm{\lambda }\right)+\mathrm{y}\left(-3+4\mathrm{\lambda }\right)-\left(4+5\mathrm{\lambda }\right)=0..\left(1\right) \\ \mathrm{This}\mathrm{is}\perp \mathrm{to}\mathrm{the}\mathrm{line}6\mathrm{x}-7\mathrm{y}+8=0 \\ \mathrm{So},-\left(\frac{2+3\mathrm{\lambda }}{-3+4\mathrm{\lambda }}\right)\left(\frac{6}{7}\right)=-1 \\ \mathrm{So},\mathrm{\lambda }=\frac{33}{10} \\ \mathrm{Put}\mathrm{in}\mathrm{eq}\left(1\right),\mathrm{we}\mathrm{get} \\ \mathrm{The}\mathrm{eq}\mathrm{of}\mathrm{the}\mathrm{required}\mathrm{line}\mathrm{is}119\mathrm{x}+102\mathrm{y}-205=0 \end{gathered}$$
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