If (a, b) is the mid-point of the line segment joining the points A(10, -6) and B(k, 4) and a-2b = 18, the value of k is:

Options

A. 30
B. 22
C. 4
D. 40

Answer

To solve the problem step by step, we need to find the value of \( k \) given that \( (a, b) \) is the midpoint of the line segment joining the points \( A(10, -6) \) and \( B(k, 4) \), and that \( a - 2b = 18 \).

### Step 1: Find the coordinates of the midpoint \( (a, b) \)
The formula for the midpoint \( (x_m, y_m) \) of a line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ (x_m, y_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, \( A(10, -6) \) and \( B(k, 4) \).

So, we can calculate:
\[ a = \frac{10 + k}{2} \]
\[ b = \frac{-6 + 4}{2} = \frac{-2}{2} = -1 \]

### Step 2: Substitute the values of \( a \) and \( b \) into the equation \( a - 2b = 18 \)
We have \( b = -1 \), so we substitute \( b \) into the equation:
\[ a - 2(-1) = 18 \]
This simplifies to:
\[ a + 2 = 18 \]
Thus, we can find \( a \):
\[ a = 18 - 2 = 16 \]

### Step 3: Substitute \( a \) back to find \( k \)
Now we have \( a = \frac{10 + k}{2} \) and we know \( a = 16 \):
\[ \frac{10 + k}{2} = 16 \]
To eliminate the fraction, multiply both sides by 2:
\[ 10 + k = 32 \]
Now, isolate \( k \):
\[ k = 32 - 10 = 22 \]

### Final Answer
The value of \( k \) is \( 22 \).

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