What is the value of 'a' if the mid-point of the line segment joining the points P(6,a-2) and Q(-2, 4) is (2,-4)?

Options

A. `-10`
B. 10
C. 0
D. 7

Answer

To find the value of 'a' such that the midpoint of the line segment joining the points P(6, a-2) and Q(-2, 4) is (2, -4), we can follow these steps:

### Step 1: Understand the midpoint formula
The midpoint M of a line segment joining two points P(x1, y1) and Q(x2, y2) is given by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

### Step 2: Identify the coordinates of points P and Q
From the problem, we have:
- Point P: \( (6, a - 2) \) where \( x_1 = 6 \) and \( y_1 = a - 2 \)
- Point Q: \( (-2, 4) \) where \( x_2 = -2 \) and \( y_2 = 4 \)

### Step 3: Set up the midpoint equation
Using the midpoint formula:
\[ M = \left( \frac{6 + (-2)}{2}, \frac{(a - 2) + 4}{2} \right) \]
This simplifies to:
\[ M = \left( \frac{4}{2}, \frac{a + 2}{2} \right) = (2, \frac{a + 2}{2}) \]

### Step 4: Set the midpoint equal to the given midpoint
We know from the problem that the midpoint M is (2, -4). Therefore, we can set up the equations:
1. \( \frac{4}{2} = 2 \) (which is already satisfied)
2. \( \frac{a + 2}{2} = -4 \)

### Step 5: Solve for 'a'
Now, we solve the second equation:
\[ \frac{a + 2}{2} = -4 \]
To eliminate the fraction, multiply both sides by 2:
\[ a + 2 = -8 \]
Now, subtract 2 from both sides:
\[ a = -8 - 2 \]
\[ a = -10 \]

### Conclusion
The value of 'a' is \( -10 \).

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