Find the equations of the st.lines which pass through `(4,5)` and make angle `45^(@)` with the st.line `2x+y+1=0`

Answer

To find the equations of the straight lines that pass through the point (4, 5) and make an angle of 45 degrees with the line given by the equation \(2x + y + 1 = 0\), we can follow these steps:

### Step 1: Find the slope of the given line
The equation of the line is in the form \(Ax + By + C = 0\). Here, \(A = 2\), \(B = 1\), and \(C = 1\).

The slope \(M_1\) of the line can be calculated using the formula:
\[ M_1 = -\frac{A}{B} = -\frac{2}{1} = -2 \]

### Step 2: Use the angle formula
We know that if two lines have slopes \(M_1\) and \(M_2\), the angle \(\theta\) between them can be given by the formula:
\[ \tan(\theta) = \left| \frac{M_1 - M_2}{1 + M_1M_2} \right| \]
Given that \(\theta = 45^\circ\), we have \(\tan(45^\circ) = 1\). Thus, we can set up the equation:
\[ 1 = \left| \frac{-2 - M_2}{1 + (-2)M_2} \right| \]

### Step 3: Solve the equation
This absolute value equation gives us two cases to consider.

**Case 1:**
\[ 1 = \frac{-2 - M_2}{1 - 2M_2} \]
Cross-multiplying gives:
\[ 1 - 2M_2 = -2 - M_2 \]
Rearranging terms:
\[ 1 + 2 = -M_2 + 2M_2 \implies 3 = M_2 \implies M_2 = 3 \]

**Case 2:**
\[ 1 = \frac{2 + M_2}{1 - 2M_2} \]
Cross-multiplying gives:
\[ 1 - 2M_2 = 2 + M_2 \]
Rearranging terms:
\[ 1 - 2 = M_2 + 2M_2 \implies -1 = 3M_2 \implies M_2 = -\frac{1}{3} \]

### Step 4: Write the equations of the lines
Now we have two slopes \(M_2 = 3\) and \(M_2 = -\frac{1}{3}\).

Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1) = (4, 5)\).

**For \(M_2 = 3\):**
\[ y - 5 = 3(x - 4) \]
Expanding this:
\[ y - 5 = 3x - 12 \implies y = 3x - 7 \]

**For \(M_2 = -\frac{1}{3}\):**
\[ y - 5 = -\frac{1}{3}(x - 4) \]
Expanding this:
\[ y - 5 = -\frac{1}{3}x + \frac{4}{3} \implies y = -\frac{1}{3}x + \frac{4}{3} + 5 \]
Converting \(5\) to a fraction:
\[ y = -\frac{1}{3}x + \frac{4}{3} + \frac{15}{3} = -\frac{1}{3}x + \frac{19}{3} \]

### Final Equations
The equations of the two straight lines are:
1. \(y = 3x - 7\)
2. \(y = -\frac{1}{3}x + \frac{19}{3}\)