If the points A(x,2), B(-3, -4) and C(7, -5) are collinear, then the value of x is:

Options

A. -63
B. 63
C. -60
D. 60

Answer

To determine the value of \( x \) such that the points \( A(x, 2) \), \( B(-3, -4) \), and \( C(7, -5) \) are collinear, we can use the condition for collinearity of three points in the coordinate plane. The formula for checking the collinearity of points \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by:

\[ \Delta = x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) = 0 \]

### Step 1: Identify the coordinates
Here, we have:
- \( A(x, 2) \) where \( x_1 = x \) and \( y_1 = 2 \)
- \( B(-3, -4) \) where \( x_2 = -3 \) and \( y_2 = -4 \)
- \( C(7, -5) \) where \( x_3 = 7 \) and \( y_3 = -5 \)

### Step 2: Substitute the coordinates into the collinearity formula
Substituting the values into the formula, we get:

\[ x( -4 - (-5) ) + (-3)( -5 - 2 ) + 7( 2 - (-4) ) = 0 \]

### Step 3: Simplify the equation
Now, simplify each term:
- The first term becomes \( x(-4 + 5) = x(1) = x \)
- The second term becomes \( -3(-5 - 2) = -3(-7) = 21 \)
- The third term becomes \( 7(2 + 4) = 7(6) = 42 \)

Putting it all together, we have:

\[ x + 21 + 42 = 0 \]

### Step 4: Combine like terms
Combine the constants:

\[ x + 63 = 0 \]

### Step 5: Solve for \( x \)
Now, isolate \( x \):

\[ x = -63 \]

Thus, the value of \( x \) is \( -63 \).

### Final Answer:
The value of \( x \) is \( -63 \).

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