Point P divides the line segment joining the points A (8,0) and B (16, -8) in the ratio 3:5. Find its co-ordinates of point P.
Also, find the equation of the line through P and parallel to `3x + 5y = 7`. 

Answer

To solve the problem step by step, we will first find the coordinates of point P that divides the line segment joining points A (8, 0) and B (16, -8) in the ratio 3:5. Then, we will find the equation of the line through point P that is parallel to the line given by the equation \(3x + 5y = 7\).

### Step 1: Use the Section Formula to Find Coordinates of Point P

The section formula states that if a point P 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 point P \((x, y)\) can be calculated as follows:

\[ x = \frac{Mx_2 + Nx_1}{M + N} \]
\[ y = \frac{My_2 + Ny_1}{M + N} \]

Here, we have:
- \(A(8, 0)\) where \(x_1 = 8\) and \(y_1 = 0\)
- \(B(16, -8)\) where \(x_2 = 16\) and \(y_2 = -8\)
- The ratio \(M:N = 3:5\) (where \(M = 3\) and \(N = 5\))

Now, substituting the values into the formulas:

#### Calculate x-coordinate of P:

\[ x = \frac{3 \cdot 16 + 5 \cdot 8}{3 + 5} = \frac{48 + 40}{8} = \frac{88}{8} = 11 \]

#### Calculate y-coordinate of P:

\[ y = \frac{3 \cdot (-8) + 5 \cdot 0}{3 + 5} = \frac{-24 + 0}{8} = \frac{-24}{8} = -3 \]

Thus, the coordinates of point P are \((11, -3)\).

### Step 2: Find the Equation of the Line Through P Parallel to \(3x + 5y = 7\)

To find the equation of the line through point P that is parallel to the line \(3x + 5y = 7\), we first need to determine the slope of the given line.

The slope \(m\) of a line in the form \(Ax + By = C\) is given by:

\[ m = -\frac{A}{B} \]

For the line \(3x + 5y = 7\):
- \(A = 3\)
- \(B = 5\)

Thus, the slope is:

\[ m = -\frac{3}{5} \]

Since parallel lines have the same slope, the slope of the line we need to find is also \(-\frac{3}{5}\).

Now, we will use the point-slope form of the equation of a line, which is:

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

Substituting \(m = -\frac{3}{5}\) and the coordinates of point P \((11, -3)\):

\[ y - (-3) = -\frac{3}{5}(x - 11) \]

This simplifies to:

\[ y + 3 = -\frac{3}{5}x + \frac{33}{5} \]

To eliminate the fraction, we can multiply the entire equation by 5:

\[ 5y + 15 = -3x + 33 \]

Rearranging gives us:

\[ 3x + 5y + 15 - 33 = 0 \]
\[ 3x + 5y - 18 = 0 \]

Thus, the equation of the line through point P and parallel to the line \(3x + 5y = 7\) is:

\[ 3x + 5y = 18 \]

### Summary of the Solution:
- The coordinates of point P are \((11, -3)\).
- The equation of the line through P, parallel to \(3x + 5y = 7\), is \(3x + 5y = 18\).