Write down the equation of the line whose gradient is `-2/5 ` and which passes through point P, where P divides the line segment joining A (4, -8) and B (12, 0) in the ratio 3:1. 

Answer

To find the equation of the line with a gradient of \(-\frac{2}{5}\) that passes through point \(P\), which divides the line segment joining points \(A(4, -8)\) and \(B(12, 0)\) in the ratio \(3:1\), we will follow these steps:

### Step 1: Find the coordinates of point \(P\) using the section formula.
The section formula states that if a point \(P(x_1, y_1)\) divides the line segment joining points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m:n\), then the coordinates of \(P\) can be calculated as follows:

\[ x_1 = \frac{mx_2 + nx_1}{m+n} \]
\[ y_1 = \frac{my_2 + ny_1}{m+n} \]

Here, \(A(4, -8)\), \(B(12, 0)\), \(m = 3\), and \(n = 1\).

Calculating the x-coordinate of \(P\):

\[ x_1 = \frac{3 \cdot 12 + 1 \cdot 4}{3 + 1} = \frac{36 + 4}{4} = \frac{40}{4} = 10 \]

Calculating the y-coordinate of \(P\):

\[ y_1 = \frac{3 \cdot 0 + 1 \cdot (-8)}{3 + 1} = \frac{0 - 8}{4} = \frac{-8}{4} = -2 \]

So, the coordinates of point \(P\) are \(P(10, -2)\).

### Step 2: Use the point-slope form to write the equation of the line.
The point-slope form of the equation of a line is given by:

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

Where \(m\) is the slope and \((x_1, y_1)\) are the coordinates of the point through which the line passes.

Substituting \(m = -\frac{2}{5}\), \(x_1 = 10\), and \(y_1 = -2\):

\[ y - (-2) = -\frac{2}{5}(x - 10) \]

This simplifies to:

\[ y + 2 = -\frac{2}{5}(x - 10) \]

### Step 3: Rearranging the equation to standard form.
Now, we will simplify and rearrange the equation:

\[ y + 2 = -\frac{2}{5}x + 4 \]

Subtracting 2 from both sides:

\[ y = -\frac{2}{5}x + 4 - 2 \]
\[ y = -\frac{2}{5}x + 2 \]

To convert this into standard form \(Ax + By + C = 0\), we can rearrange:

\[ \frac{2}{5}x + y - 2 = 0 \]

Multiplying through by 5 to eliminate the fraction:

\[ 2x + 5y - 10 = 0 \]

### Final Equation of the Line
Thus, the required equation of the line is:

\[ 2x + 5y - 10 = 0 \]

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