The points A, B and C are (4, 0), (2, 2) and (0, 6) respectively. AB produced cuts the y-axis at P and CB produced cuts the x-axis at Q. Find the co-ordinates of the points P and Q. Find the equation of the straight line joining the mid-points of Ac and OB (where O is the origin), and verify that this line passes through the mid- point of PQ.

Answer

To solve the problem step by step, we will first find the coordinates of points P and Q, then determine the equation of the line joining the midpoints of AC and OB, and finally verify that this line passes through the midpoint of PQ.

### Step 1: Find the coordinates of point P (where AB produced cuts the y-axis)

1. **Find the slope of line AB:**
- Coordinates of A = (4, 0) and B = (2, 2)
- Slope (m) = (y2 - y1) / (x2 - x1) = (2 - 0) / (2 - 4) = 2 / -2 = -1

2. **Use point-slope form to find the equation of line AB:**
- Equation: y - y1 = m(x - x1)
- y - 0 = -1(x - 4)
- y = -x + 4

3. **Find the y-intercept (point P):**
- Set x = 0 in the equation of line AB: y = -0 + 4 = 4
- Thus, P = (0, 4)

### Step 2: Find the coordinates of point Q (where CB produced cuts the x-axis)

1. **Find the slope of line CB:**
- Coordinates of C = (0, 6) and B = (2, 2)
- Slope (m) = (2 - 6) / (2 - 0) = -4 / 2 = -2

2. **Use point-slope form to find the equation of line CB:**
- Equation: y - y1 = m(x - x1)
- y - 6 = -2(x - 0)
- y = -2x + 6

3. **Find the x-intercept (point Q):**
- Set y = 0 in the equation of line CB: 0 = -2x + 6
- 2x = 6 → x = 3
- Thus, Q = (3, 0)

### Step 3: Find the midpoints of AC and OB

1. **Midpoint of AC:**
- A = (4, 0), C = (0, 6)
- Midpoint M1 = ((x1 + x2)/2, (y1 + y2)/2) = ((4 + 0)/2, (0 + 6)/2) = (2, 3)

2. **Midpoint of OB:**
- O = (0, 0), B = (2, 2)
- Midpoint M2 = ((0 + 2)/2, (0 + 2)/2) = (1, 1)

### Step 4: Find the equation of the line joining midpoints M1 and M2

1. **Find the slope of line M1M2:**
- M1 = (2, 3), M2 = (1, 1)
- Slope (m) = (1 - 3) / (1 - 2) = -2 / -1 = 2

2. **Use point-slope form to find the equation of line M1M2:**
- Equation: y - y1 = m(x - x1)
- y - 1 = 2(x - 1)
- y - 1 = 2x - 2
- y = 2x - 1

### Step 5: Verify that the line passes through the midpoint of PQ

1. **Find the midpoint of PQ:**
- P = (0, 4), Q = (3, 0)
- Midpoint = ((0 + 3)/2, (4 + 0)/2) = (3/2, 2)

2. **Check if this point satisfies the equation of line M1M2:**
- Substitute x = 3/2 into the equation y = 2x - 1:
- y = 2(3/2) - 1 = 3 - 1 = 2
- Since the calculated y = 2 matches the y-coordinate of the midpoint (3/2, 2), the line passes through this point.

### Final Results

- Coordinates of P: (0, 4)
- Coordinates of Q: (3, 0)
- Equation of the line joining midpoints M1 and M2: y = 2x - 1
- The line passes through the midpoint of PQ: (3/2, 2)