Write down the equation of the line whose gradient is `3/2` and which passes through P, where P divides the line segment joining `A(-2, 6) and B(3,-4) ` in the ratio 2 : 3. 

Answer

To find the equation of the line with a gradient of \( \frac{3}{2} \) that passes through point \( P \), which divides the line segment joining points \( A(-2, 6) \) and \( B(3, -4) \) in the ratio \( 2:3 \), we can follow these steps:

### Step 1: Find the coordinates of point P
Point \( P \) divides the line segment \( AB \) in the ratio \( 2:3 \). We can use the section formula to find the coordinates of point \( P \).

The section formula states that if a point \( P(x, y) \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then:
\[ P_x = \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \quad P_y = \frac{m \cdot y_2 + n \cdot y_1}{m+n} \]

Here, \( A(-2, 6) \), \( B(3, -4) \), \( m = 2 \), and \( n = 3 \).

Calculating the x-coordinate of \( P \):
\[ P_x = \frac{2 \cdot 3 + 3 \cdot (-2)}{2 + 3} = \frac{6 - 6}{5} = \frac{0}{5} = 0 \]

Calculating the y-coordinate of \( P \):
\[ P_y = \frac{2 \cdot (-4) + 3 \cdot 6}{2 + 3} = \frac{-8 + 18}{5} = \frac{10}{5} = 2 \]

Thus, the coordinates of point \( P \) are \( (0, 2) \).

### Step 2: Use the point-slope form to find the equation of the line
We know the gradient (slope) \( m = \frac{3}{2} \) and the point \( P(0, 2) \). The equation of a line in point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = \frac{3}{2} \), \( x_1 = 0 \), and \( y_1 = 2 \):
\[ y - 2 = \frac{3}{2}(x - 0) \]
This simplifies to:
\[ y - 2 = \frac{3}{2}x \]

### Step 3: Rearranging to the slope-intercept form
To express the equation in the slope-intercept form \( y = mx + c \):
\[ y = \frac{3}{2}x + 2 \]

### Step 4: Rearranging to standard form (if needed)
If we want to express this in standard form \( Ax + By + C = 0 \):
\[ 2y = 3x + 4 \quad \Rightarrow \quad 3x - 2y + 4 = 0 \]

Thus, the required equation of the line is:
\[ 3x - 2y + 4 = 0 \]