Question

If the points \((2, 3)\), \((0, 4)\) and \((p, 2)\) are collinear, then the value of \(p\) is:

• A. \(\frac{4}{7}\)
• B. \(-\frac{3}{7}\)
• C. \(4\)
• D. \(-4\)

Hint:

Equating slopes (\(\frac{y_2 - y_1}{x_2 - x_1} = \frac{y_3 - y_2}{x_3 - x_2}\)) is generally much faster and less prone to simple sign calculation errors than working out the complete determinant formula for a triangle's area.

Answer 1

The Correct Option is C

Concept: Three points \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\) are said to be collinear if they lie along the exact same straight line. This geometric property can be solved mathematically using two primary methods:
• Slope Method: The slope of line segment \(AB\) must equal the slope of line segment \(BC\).
• Area of Triangle Method: The area of the triangle formed by three collinear points must equal zero.

Step 1: Apply the Slope Method
Let the given coordinate points be labeled as follows: \[ A = (2, 3), \quad B = (0, 4), \quad C = (p, 2) \] The formula for the slope \(m\) passing through any two coordinate pairs \((x_a, y_a)\) and \((x_b, y_b)\) is given by: \[ m = \frac{y_b - y_a}{x_b - x_a} \] Let us compute the slope of the line segment \(AB\): \[ m_{AB} = \frac{4 - 3}{0 - 2} = \frac{1}{-2} = -\frac{1}{2} \] Now, let us compute the slope of the line segment \(BC\): \[ m_{BC} = \frac{2 - 4}{p - 0} = \frac{-2}{p} \]

Step 2: Equate the slopes to find \(p\)
Since the points are collinear, the slopes must be perfectly identical (\(m_{AB} = m_{BC}\)): \[ -\frac{1}{2} = \frac{-2}{p} \] Eliminating the negative signs from both sides of the linear relation: \[ \frac{1}{2} = \frac{2}{p} \] Cross-multiplying to solve for the variable \(p\): \[ 1 \times p = 2 \times 2 \quad \Rightarrow \quad p = 4 \]

Step 3: Verification using Area of Triangle formula
Let us double-check using the determinant area method, where \(\Delta = 0\): \[ \frac{1}{2} [x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)] = 0 \] Substituting the coordinates: \[ 2(4 - 2) + 0(2 - 3) + p(3 - 4) = 0 \] \[ 2(2) + 0 + p(-1) = 0 \] \[ 4 - p = 0 \quad \Rightarrow \quad p = 4 \] Both independent approaches confirm that \(p = 4\).