The equations of the lines through the point $(3, 2)$ which makes an angle of $45 °$ with the line $x - 2 y = 3$ are

$3 x - y = 7 & x + 3 y = 9$

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$x - 3 y = 7 & 3 x + y = 9$

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$x - y = 3 & x + y = 2$

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$2 x + y = 7 & x - 2 y = 9$

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Solution

The correct option is A
$3 x - y = 7 & x + 3 y = 9$

Step 1: Solve for the slope of line.
The given equation of line is $x - 2 y = 3$ . $\Rightarrow y = \frac{x}{2} - \frac{3}{2}$
It is of the slope intercept form of a line i.e. $y = m x + c$ where $m$ is the slope and $c$ is the y-intercept
$∴$ Slope of the line, $m_{1} = \frac{1}{2}$ .
Angle between two lines is given by $\tan (θ) = |\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}| \Rightarrow \tan (θ) = \{\begin{cases} + (\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}) & \forall m_{2} > m_{1} \\ - (\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}) & \forall m_{2} < m_{1} \end{cases}$
Here it is given that the angle between two lines is $45 °$
Case 1: $m_{2} > m_{1}$ .
$\begin{array}{rcl} ∴ \tan (45 °) & = & \frac{(m_{2} - \frac{1}{2})}{1 + \frac{m_{2}}{2}} \end{array}$
$\Rightarrow$ $\begin{array}{rcl} 1 & = & \frac{(2 m_{2} - 1)}{2 + m_{2}} \end{array}$
$\Rightarrow$ $\begin{array}{rcl} 2 + m_{2} & = & 2 m_{2} - 1 \end{array}$
$\Rightarrow$ $\begin{array}{rcl} m_{2} & = & 3 \end{array}$
Case 2: $m_{2} < m_{1}$
$\begin{array}{rcl} ∴ \tan (45 °) & = & \frac{- (m_{2} - \frac{1}{2})}{1 + \frac{m_{2}}{2}} \end{array}$
$\Rightarrow$ $\begin{array}{rcl} 1 & = & \frac{- (2 m_{2} - 1)}{2 + m_{2}} \end{array}$
$\Rightarrow$ $\begin{array}{rcl} 2 + m_{2} & = & - 2 m_{2} + 1 \end{array}$
$\Rightarrow$ $\begin{array}{rcl} m_{2} & = & \frac{- 1}{3} \end{array}$
Step 2: Solve for the equation of line.
Equation of line passing through $(x_{1}, y_{1})$ is $(y - y_{1}) = m (x - x_{1})$
Equation of line passing through $(3, 2)$ with slope $\begin{array}{rcl} m_{2} & = & 3 \end{array}$ is
$(y - 2) = 3 (x - 3) \Rightarrow 3 x - y - 7 = 0$
Equation of line passing through $(3, 2)$ with slope $\begin{array}{rcl} m_{2} & = & \frac{- 1}{3} \end{array}$ is
$(y - 2) = \frac{- 1}{3} (x - 3) \Rightarrow x + 3 y - 9 = 0$
Hence, option(A) is correct i.e. $3 x - y = 7 & x + 3 y = 9$

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