Question

If the points (x, 9), (0, 1) and (-6,-7) are collinear, then the value of x is

• A. 4
• B. 5
• C. 6
• D. 7

Answer 1

The Correct Option is C

Step 1: Recall the condition for collinearity of three points.

Three points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are collinear if the area of the triangle formed by them is zero. The formula for the area of a triangle is:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|. \]

For collinearity, the area must be zero:

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

Step 2: Substitute the coordinates of the given points.

The points are \( (x, 9) \), \( (0, 1) \), and \( (-6, -7) \). Substituting into the collinearity condition:

\[ x(1 - (-7)) + 0((-7) - 9) + (-6)(9 - 1) = 0. \]

Simplify each term:

\[ x(1 + 7) + 0(-16) + (-6)(8) = 0. \]

\[ x(8) + 0 - 48 = 0. \]

Step 3: Solve for \( x \).

\[ 8x - 48 = 0 \implies 8x = 48 \implies x = 6. \]

Final Answer: The value of \( x \) is \( \mathbf{6} \), which corresponds to option \( \mathbf{(3)} \).