The points `(x, 2x), (2y, y)` and (3, 3) are collinear

Options

A. for all values of (x, y)
B. 2 is AM of xy
C. 2 is GM of x, y
D. 2 is HM of x, y

Answer

To determine if the points \((x, 2x)\), \((2y, y)\), and \((3, 3)\) are collinear, we can use the concept of slopes. If the slopes between any two pairs of points are equal, then the points are collinear.

### Step-by-Step Solution:

1. **Identify the Points**: The points given are:
- Point A: \((x, 2x)\)
- Point B: \((2y, y)\)
- Point C: \((3, 3)\)

2. **Calculate the Slope between Points A and B**:
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For points A and B:
\[ m_{AB} = \frac{y - 2x}{2y - x} \]

3. **Calculate the Slope between Points B and C**:
For points B and C:
\[ m_{BC} = \frac{3 - y}{3 - 2y} \]

4. **Set the Slopes Equal**:
Since the points are collinear, we set the slopes equal:
\[ \frac{y - 2x}{2y - x} = \frac{3 - y}{3 - 2y} \]

5. **Cross Multiply**:
Cross multiplying gives us:
\[ (y - 2x)(3 - 2y) = (3 - y)(2y - x) \]

6. **Expand Both Sides**:
Expanding both sides:
\[ 3y - 2y^2 - 6x + 4xy = 6y - 3x - 2y^2 + xy \]

7. **Rearrange the Equation**:
Bringing all terms to one side:
\[ 3y - 6y - 6x + 3x + 4xy - xy = 0 \]
Simplifying gives:
\[ -3y - 3x + 3xy = 0 \]

8. **Factor Out Common Terms**:
Factoring out \(-3\):
\[ -3(y + x - xy) = 0 \]
Since \(-3 \neq 0\), we can set the remaining expression to zero:
\[ y + x - xy = 0 \]

9. **Rearrange to Find Relationship**:
Rearranging gives:
\[ xy = x + y \]

10. **Multiply Both Sides by 2**:
Multiplying both sides by 2 results in:
\[ 2 = \frac{2xy}{x + y} \]
This shows that 2 is the harmonic mean of \(x\) and \(y\).

### Conclusion:
Thus, we conclude that the points \((x, 2x)\), \((2y, y)\), and \((3, 3)\) are collinear if \(2\) is the harmonic mean of \(x\) and \(y\).