The equation of a line passing through the point of intersection of line `2x+3y +1=0 and 3x- 5y -5=0` and making an angle of `45^(@)` with positive X-axis is

Options

A. `2x -19y +23=0`
B. `19x-23y+15=0`
C. `19x-19y-23=0`
D. `20x -19y +23 =0`

Answer

To solve the problem, we need to find the equation of a line that passes through the point of intersection of the two given lines and makes an angle of \(45^\circ\) with the positive x-axis. Let's break this down step by step.

### Step 1: Find the point of intersection of the two lines

The equations of the lines are:
1. \(2x + 3y + 1 = 0\) (Equation 1)
2. \(3x - 5y - 5 = 0\) (Equation 2)

To find the intersection, we can solve these equations simultaneously.

From Equation 1, we can express \(y\) in terms of \(x\):
\[ 3y = -2x - 1 \implies y = -\frac{2}{3}x - \frac{1}{3} \]

Now, substitute this expression for \(y\) into Equation 2:
\[ 3x - 5\left(-\frac{2}{3}x - \frac{1}{3}\right) - 5 = 0 \]
\[ 3x + \frac{10}{3}x + \frac{5}{3} - 5 = 0 \]
Multiply through by 3 to eliminate the fraction:
\[ 9x + 10x + 5 - 15 = 0 \]
\[ 19x - 10 = 0 \implies x = \frac{10}{19} \]

Now substitute \(x\) back into the expression for \(y\):
\[ y = -\frac{2}{3}\left(\frac{10}{19}\right) - \frac{1}{3} = -\frac{20}{57} - \frac{19}{57} = -\frac{39}{57} = -\frac{13}{19} \]

Thus, the point of intersection is:
\[ \left(\frac{10}{19}, -\frac{13}{19}\right) \]

### Step 2: Determine the slope of the line making a \(45^\circ\) angle with the x-axis

The slope \(m\) of a line making an angle \(\theta\) with the positive x-axis is given by:
\[ m = \tan(\theta) \]
For \(\theta = 45^\circ\):
\[ m = \tan(45^\circ) = 1 \]

### Step 3: Use the point-slope form to find the equation of the line

The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Substituting \(m = 1\) and the point \(\left(\frac{10}{19}, -\frac{13}{19}\right)\):
\[ y - \left(-\frac{13}{19}\right) = 1\left(x - \frac{10}{19}\right) \]
\[ y + \frac{13}{19} = x - \frac{10}{19} \]
Rearranging gives:
\[ y = x - \frac{10}{19} - \frac{13}{19} \]
\[ y = x - \frac{23}{19} \]

### Step 4: Convert to standard form

To convert this to standard form \(Ax + By + C = 0\):
\[ y - x + \frac{23}{19} = 0 \]
Multiplying through by 19 to eliminate the fraction:
\[ 19y - 19x + 23 = 0 \]
Rearranging gives:
\[ 19x - 19y - 23 = 0 \]

### Final Answer

The equation of the line is:
\[ 19x - 19y - 23 = 0 \]

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