Question

The equations of the lines through the point (3, 2) which make an angle of 45^o with the line x - 2y = 3 are

• (A) 3x - y = 7 and x + 3y = 9
• (B) x - 3y = 7 and 3x + y = 9
• (C) x - y = 3 and x + y = 2
• (D) 2x + y = 7 and x - 2y = 9

Answer 1

The Correct Option is A

Explanation:
The slope of line $x-2y=3$ is $\frac{1}{2}$Let the slope of required lines is $m$$$\begin{array}{lr}\therefore & \tan {45}^{\circ }=\pm |\frac{2}{2}-m|\\ \Rightarrow 1+\frac{m}{2}=\pm (\frac{1}{2}-m)& \frac{m}{2}\mid \\ \Rightarrow -\frac{1}{2}=\frac{3m}{2}\Rightarrow m=\frac{-1}{3}\end{array}$$or $1+\frac{m}{2}=\frac{-1}{2}+m$$$\Rightarrow \frac{m}{2}=\frac{3}{2}\Rightarrow m=3$$$\therefore$ Equation of line with slope $m=\frac{1}{-3}$ and passing through $(3,2),$ is$$\begin{array}{l}(y-2)=\frac{1}{-3}(x-3)\\ \Rightarrow y-6=-x+3\end{array}$$$\Rightarrow x+3y=9$And another equation of line with slope$m=3$ and passing through $(3,2),$ is$$(y-2)=3(x-3)$$$\Rightarrow y-2=3x-9$$\Rightarrow 3x-y=7$