Find the equation of the lines through the point $(3, 2)$ which make an angle of $45^\circ$ with the line $x - 2y = 3$.

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Accepted Answer

> [!TLDR]
> We use the slope formula for the angle between two lines. The slopes of the required lines are $m = 3$ and $m = -\frac{1}{3}$, giving the equations $3x - y = 7$ and $x + 3y = 9$.

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The key idea: when a line makes a given angle with another line, the slopes are related by the tangent of the angle between them. Here, the angle is $45^\circ$, so $	an 45^\circ = 1$. That gives a clean equation linking the unknown slope $m$ to the known slope of the given line.

First, find the slope of the given line $x - 2y = 3$. Rewrite it as $y = \frac{1}{2}x - \frac{3}{2}$, so its slope is $m_1 = \frac{1}{2}$.

Now, if a line with slope $m$ makes an angle $	heta$ with a line of slope $m_1$, then:

> [!FORMULA]
> $$ 	an 	heta = \left| \frac{m - m_1}{1 + m m_1} ight| $$

Here $	heta = 45^\circ$, so $	an 45^\circ = 1$. Therefore:

$$ \left| \frac{m - \frac{1}{2}}{1 + m \cdot \frac{1}{2}} ight| = 1 $$

This absolute value equation gives two cases — one for the positive, one for the negative.

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**Step-by-step solution:**

1. **Set up the equation without the absolute value.**  
   $$ \frac{m - \frac{1}{2}}{1 + \frac{m}{2}} = \pm 1 $$

2. **Case 1: positive sign.**  
   $$ \frac{m - \frac{1}{2}}{1 + \frac{m}{2}} = 1 $$  
   Multiply numerator and denominator:  
   $$ m - \frac{1}{2} = 1 + \frac{m}{2} $$  
   Multiply through by 2:  
   $$ 2m - 1 = 2 + m $$  
   So $m = 3$.

3. **Case 2: negative sign.**  
   $$ \frac{m - \frac{1}{2}}{1 + \frac{m}{2}} = -1 $$  
   Multiply:  
   $$ m - \frac{1}{2} = -1 - \frac{m}{2} $$  
   Multiply by 2:  
   $$ 2m - 1 = -2 - m $$  
   So $3m = -1$, giving $m = -\frac{1}{3}$.

4. **Find the equations through $(3, 2)$.**  
   For $m = 3$:  
   $$ y - 2 = 3(x - 3) \quad\Rightarrow\quad y = 3x - 7 $$  
   Or $3x - y = 7$. …

Slope value	What the line does
$m > 0$	Rises left to right (uphill)
$m < 0$	Falls left to right (downhill)
$m = 0$	Horizontal (flat)
$m$ undefined	Vertical (straight up/down)

Q.Find the equation of the lines through the point $(3, 2)$ which make an angle of $45^\circ$ with the line $x - 2y = 3$ .

Concept understanding — Slope Calculation

Slope Calculation — From Intuition to Precision

The Intuition: Rise Over Run

The Precise Definition

What the Number Tells You

A Worked Example

Q.Find the equation of the lines through the point (3,2)(3, 2)(3,2) which make an angle of 45∘45^\circ45∘ with the line x−2y=3x - 2y = 3x−2y=3.

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Q.Find the equation of the lines through the point $(3, 2)$ which make an angle of $45^\circ$ with the line $x - 2y = 3$ .

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