C is the mid-point of PQ, if P is (4, x), C is (y, -1) and Q is (-2, 4), then x and y respectively are

Options

A. - 6 and 1
B. - 6 and 2
C. 6 and - 1
D. 6 and - 2

Answer

To solve the problem, we need to find the values of \(x\) and \(y\) given that \(C\) is the midpoint of segment \(PQ\). The coordinates of the points are as follows:

- \(P(4, x)\)
- \(C(y, -1)\)
- \(Q(-2, 4)\)

### Step 1: Use the midpoint formula for the x-coordinates
The x-coordinate of the midpoint \(C\) can be calculated using the formula:

\[ C_x = \frac{P_x + Q_x}{2} \]

Substituting the known values:

\[ y = \frac{4 + (-2)}{2} \]

### Step 2: Simplify the equation for \(y\)
Now, simplify the right-hand side:

\[ y = \frac{4 - 2}{2} = \frac{2}{2} = 1 \]

So, we have:

\[ y = 1 \]

### Step 3: Use the midpoint formula for the y-coordinates
Next, we apply the midpoint formula for the y-coordinates:

\[ C_y = \frac{P_y + Q_y}{2} \]

Substituting the known values:

\[ -1 = \frac{x + 4}{2} \]

### Step 4: Solve for \(x\)
Now, we will solve for \(x\) by multiplying both sides by 2:

\[ -2 = x + 4 \]

Now, isolate \(x\):

\[ x = -2 - 4 = -6 \]

### Final Values
Thus, the values of \(x\) and \(y\) are:

\[ x = -6, \quad y = 1 \]

### Conclusion
The final answer is \(x = -6\) and \(y = 1\).

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