Question

Equation of the straight line making equal angles with the straight lines x + y = 1 and 2x + 3y = 1 and passing through the point (1, 2) may be given as

• A. (x + y - 3) = (2x + 3y - 8)
• B. (x + y + 1) = (2x + 3y + 6)
• C. (x + y + 5) = (2x + 3y + 9)
• D. None of the above

Answer 1

The correct option is A (x + y - 3) = (2x + 3y - 8)

Equation of angle bisectors are
$\frac{x+y-1}{\sqrt{2}}=\pm \frac{2x+3y-1}{\sqrt{13}}Slope,m=\frac{\sqrt{13}-2\sqrt{2}}{3\sqrt{2}-\sqrt{13}}and-\left(\frac{\sqrt{13}+2\sqrt{2}}{3\sqrt{2}+\sqrt{13}}\right)(y-2)=\frac{\sqrt{13}-2\sqrt{2}}{3\sqrt{2}-\sqrt{13}}(x-2)\Rightarrow (3\sqrt{2}-\sqrt{13})y-6\sqrt{2}+2\sqrt{13}=(\sqrt{13}-2\sqrt{2})x-\sqrt{13}+2\sqrt{2}andy-2=-\left(\frac{\sqrt{13}+2\sqrt{2}}{3\sqrt{2}+\sqrt{13}}\right)(x-1)\Rightarrow (3\sqrt{2}-\sqrt{13})y=-6\sqrt{2}-2\sqrt{13}=-(\sqrt{13}+2)x+\sqrt{13}+2\sqrt{2}\Rightarrow \sqrt{13}(x+y-3)=-\sqrt{2}(2x+3y-8)$