The line segment joining the points (5, -4) and (2, 2) is divided by the point Q in the ratio 1:2. Does the line `x - 2y = 0` contain Q? 

Answer

To solve the problem step by step, we need to find the coordinates of point Q that divides the line segment joining the points (5, -4) and (2, 2) in the ratio 1:2, and then check if this point lies on the line given by the equation \( x - 2y = 0 \).

### Step 1: Identify the points and the ratio
Let the points be:
- \( A(5, -4) \)
- \( B(2, 2) \)

The ratio in which point Q divides the segment AB is \( 1:2 \). Here, \( m = 1 \) and \( n = 2 \).

### Step 2: Use the section formula to find the coordinates of Q
The section formula states that if a point Q divides the line segment joining points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \), then the coordinates of Q are given by:

\[ Q\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

Substituting the values:
- \( x_1 = 5, y_1 = -4 \)
- \( x_2 = 2, y_2 = 2 \)
- \( m = 1, n = 2 \)

Calculating the x-coordinate of Q:
\[ Q_x = \frac{1 \cdot 2 + 2 \cdot 5}{1 + 2} = \frac{2 + 10}{3} = \frac{12}{3} = 4 \]

Calculating the y-coordinate of Q:
\[ Q_y = \frac{1 \cdot 2 + 2 \cdot (-4)}{1 + 2} = \frac{2 - 8}{3} = \frac{-6}{3} = -2 \]

Thus, the coordinates of point Q are \( Q(4, -2) \).

### Step 3: Check if point Q lies on the line \( x - 2y = 0 \)
To determine if point Q lies on the line, we substitute the coordinates of Q into the equation of the line:

\[ x - 2y = 0 \]

Substituting \( Q(4, -2) \):
\[ 4 - 2(-2) = 4 + 4 = 8 \]

Since \( 8 \neq 0 \), point Q does not satisfy the equation of the line.

### Conclusion
Therefore, point Q does not lie on the line \( x - 2y = 0 \).

### Final Answer
No, the line \( x - 2y = 0 \) does not contain point Q.

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