Find the equation of the line which is perpendicular to the line 3x+2y=8 (4,-2) and passes through the midpoint of the line joining the points (6,4) and (4,-2)

Explanation

To find the equation of the line that is perpendicular to the line given by \(3x + 2y = 8\) and passes through the midpoint of the line segment joining the points \((6, 4)\) and \((4, -2)\), we can follow these steps:

### Step 1: Find the midpoint of the line segment joining the points \((6, 4)\) and \((4, -2)\).

The formula for the midpoint \(M\) of a line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Substituting the given points:

\[ M = \left( \frac{6 + 4}{2}, \frac{4 + (-2)}{2} \right) = \left( \frac{10}{2}, \frac{2}{2} \right) = (5, 1) \]

### Step 2: Determine the slope of the given line \(3x + 2y = 8\).

To find the slope of the line, we can rewrite the equation in slope-intercept form \(y = mx + b\):

Starting from:

\[ 3x + 2y = 8 \]

Rearranging gives:

\[ 2y = -3x + 8 \quad \Rightarrow \quad y = -\frac{3}{2}x + 4 \]

Thus, the slope \(m_1\) of the line is \(-\frac{3}{2}\).

### Step 3: Find the slope of the line that is perpendicular to the given line.

The slope \(m_2\) of the line that is perpendicular to another line is given by the negative reciprocal of the original slope:

\[ m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{3}{2}} = \frac{2}{3} \]

### Step 4: Use the point-slope form to find the equation of the required line.

We can use the point-slope form of the equation of a line, which is given by:

\[ y - y_1 = m(x - x_1) \]

Here, \((x_1, y_1) = (5, 1)\) and \(m = \frac{2}{3}\):

\[ y - 1 = \frac{2}{3}(x - 5) \]

### Step 5: Simplify the equation.

Distributing the slope on the right side:

\[ y - 1 = \frac{2}{3}x - \frac{10}{3} \]

Adding 1 to both sides:

\[ y = \frac{2}{3}x - \frac{10}{3} + 1 \]

Converting 1 to a fraction with a denominator of 3:

\[ y = \frac{2}{3}x - \frac{10}{3} + \frac{3}{3} \]

Combining the constant terms:

\[ y = \frac{2}{3}x - \frac{7}{3} \]

### Step 6: Convert to standard form.

To convert the equation to standard form \(Ax + By + C = 0\):

\[ \frac{2}{3}x - y - \frac{7}{3} = 0 \]

Multiplying through by 3 to eliminate the fractions:

\[ 2x - 3y - 7 = 0 \]

Thus, the equation of the required line is:

\[ 2x - 3y - 7 = 0 \]