The slope of a line joining P(6, k) and Q(1 - 3k, 3) is `1/(2)`. Find :
(i) k

Answer

To find the value of \( k \) given that the slope of the line joining the points \( P(6, k) \) and \( Q(1 - 3k, 3) \) is \( \frac{1}{2} \), we can follow these steps:

### Step 1: Write the formula for the slope
The formula for the slope \( m \) of a line joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

### Step 2: Assign the coordinates
For our points:
- \( P(6, k) \) gives us \( x_1 = 6 \) and \( y_1 = k \)
- \( Q(1 - 3k, 3) \) gives us \( x_2 = 1 - 3k \) and \( y_2 = 3 \)

### Step 3: Substitute into the slope formula
Substituting the values into the slope formula, we have:
\[ \frac{3 - k}{(1 - 3k) - 6} = \frac{1}{2} \]

### Step 4: Simplify the denominator
The denominator simplifies as follows:
\[ (1 - 3k) - 6 = 1 - 3k - 6 = -3k - 5 \]
Thus, the equation becomes:
\[ \frac{3 - k}{-3k - 5} = \frac{1}{2} \]

### Step 5: Cross-multiply
Cross-multiplying gives us:
\[ 2(3 - k) = 1(-3k - 5) \]

### Step 6: Expand both sides
Expanding both sides results in:
\[ 6 - 2k = -3k - 5 \]

### Step 7: Rearrange the equation
Now, rearranging the equation to isolate \( k \):
\[ 6 + 5 = -3k + 2k \]
This simplifies to:
\[ 11 = -k \]

### Step 8: Solve for \( k \)
Thus, we find:
\[ k = -11 \]

### Final Answer
The value of \( k \) is \( -11 \).

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