The line through A(4, 7) with gradient m meets the x-axis at P and the y-axis at R. The line through B(8, 3) with gradient `(-1)/(m)` meets the x-axis at Q and the y-axis at S. Find in term of m, the co-ordinates of P, Q, R and S. Obtain expressions for OP. OQ and OR. OS, where O is the point (0, 0).

Answer

To solve the problem step by step, we will find the coordinates of points P, Q, R, and S in terms of the gradient \( m \) and then derive the expressions for \( OP \), \( OQ \), \( OR \), and \( OS \).

### Step 1: Find the coordinates of point P
The line through point A(4, 7) with gradient \( m \) can be expressed using the point-slope form of the line equation:
\[ y - 7 = m(x - 4) \]
To find point P where the line meets the x-axis, we set \( y = 0 \):
\[ 0 - 7 = m(x - 4) \]
This simplifies to:
\[ -7 = m(x - 4) \]
Solving for \( x \):
\[ x - 4 = -\frac{7}{m} \implies x = 4 - \frac{7}{m} \]
Thus, the coordinates of point P are:
\[ P\left(4 - \frac{7}{m}, 0\right) \]

### Step 2: Find the coordinates of point R
To find point R where the line meets the y-axis, we set \( x = 0 \):
\[ y - 7 = m(0 - 4) \]
This simplifies to:
\[ y - 7 = -4m \implies y = 7 - 4m \]
Thus, the coordinates of point R are:
\[ R\left(0, 7 - 4m\right) \]

### Step 3: Find the coordinates of point Q
The line through point B(8, 3) with gradient \( -\frac{1}{m} \) can be expressed as:
\[ y - 3 = -\frac{1}{m}(x - 8) \]
To find point Q where the line meets the x-axis, we set \( y = 0 \):
\[ 0 - 3 = -\frac{1}{m}(x - 8) \]
This simplifies to:
\[ -3 = -\frac{1}{m}(x - 8) \implies 3m = x - 8 \implies x = 3m + 8 \]
Thus, the coordinates of point Q are:
\[ Q\left(3m + 8, 0\right) \]

### Step 4: Find the coordinates of point S
To find point S where the line meets the y-axis, we set \( x = 0 \):
\[ y - 3 = -\frac{1}{m}(0 - 8) \]
This simplifies to:
\[ y - 3 = \frac{8}{m} \implies y = 3 + \frac{8}{m} \]
Thus, the coordinates of point S are:
\[ S\left(0, 3 + \frac{8}{m}\right) \]

### Step 5: Obtain expressions for OP, OQ, OR, and OS
Now we will find the distances from the origin O(0, 0) to points P, Q, R, and S.

1. **Distance OP**:
\[ OP = \sqrt{\left(4 - \frac{7}{m}\right)^2 + 0^2} = \left|4 - \frac{7}{m}\right| \]

2. **Distance OQ**:
\[ OQ = \sqrt{\left(3m + 8\right)^2 + 0^2} = |3m + 8| \]

3. **Distance OR**:
\[ OR = \sqrt{0^2 + \left(7 - 4m\right)^2} = |7 - 4m| \]

4. **Distance OS**:
\[ OS = \sqrt{0^2 + \left(3 + \frac{8}{m}\right)^2} = \left|3 + \frac{8}{m}\right| \]

### Summary of Coordinates:
- \( P\left(4 - \frac{7}{m}, 0\right) \)
- \( Q\left(3m + 8, 0\right) \)
- \( R\left(0, 7 - 4m\right) \)
- \( S\left(0, 3 + \frac{8}{m}\right) \)

### Summary of Distances:
- \( OP = \left|4 - \frac{7}{m}\right| \)
- \( OQ = |3m + 8| \)
- \( OR = |7 - 4m| \)
- \( OS = \left|3 + \frac{8}{m}\right| \)