The line `4x - 3y + 12 = 0 ` meets x-axis at A. Write the co-ordinates of A. Determine the equation of the line through A and perpendicular to `4x - 3y + 12 = 0`. 

Answer

To solve the problem step by step, we will first find the coordinates of point A where the line meets the x-axis, and then we will determine the equation of the line that passes through point A and is perpendicular to the given line.

### Step 1: Find the coordinates of point A
The line is given by the equation:
\[ 4x - 3y + 12 = 0 \]

To find the point where this line meets the x-axis, we set \( y = 0 \) (since any point on the x-axis has a y-coordinate of 0).

Substituting \( y = 0 \) into the equation:
\[ 4x - 3(0) + 12 = 0 \]
This simplifies to:
\[ 4x + 12 = 0 \]

Now, solve for \( x \):
\[ 4x = -12 \]
\[ x = -3 \]

Thus, the coordinates of point A are:
\[ A(-3, 0) \]

### Step 2: Determine the slope of the given line
The slope-intercept form of a line is given by \( y = mx + b \), where \( m \) is the slope. We can rearrange the given line equation to find its slope.

Starting from:
\[ 4x - 3y + 12 = 0 \]
Rearranging gives:
\[ -3y = -4x - 12 \]
\[ y = \frac{4}{3}x + 4 \]

Thus, the slope of the given line is:
\[ m_1 = \frac{4}{3} \]

### Step 3: Find the slope of the perpendicular line
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, if the slope of the given line is \( \frac{4}{3} \), the slope \( m_2 \) of the perpendicular line is:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{4}{3}} = -\frac{3}{4} \]

### Step 4: Write the equation of the perpendicular line
We now have the slope of the perpendicular line \( m_2 = -\frac{3}{4} \) and a point \( A(-3, 0) \) through which it passes.

Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (-3, 0) \) and \( m = -\frac{3}{4} \):
\[ y - 0 = -\frac{3}{4}(x + 3) \]

Expanding this:
\[ y = -\frac{3}{4}x - \frac{3}{4} \cdot 3 \]
\[ y = -\frac{3}{4}x - \frac{9}{4} \]

To express this in standard form \( Ax + By + C = 0 \), we can multiply through by 4 to eliminate the fraction:
\[ 4y = -3x - 9 \]
Rearranging gives:
\[ 3x + 4y + 9 = 0 \]

### Final Answer
The coordinates of point A are:
\[ A(-3, 0) \]
The equation of the line through A and perpendicular to the given line is:
\[ 3x + 4y + 9 = 0 \]