Question

If the points P(2, 3), Q(5, k) and R(6, 7) are collinear, then the value of k is

• A. 4
• B. \(\frac{1}{4}\)
• C. \(-\frac{3}{4}\)
• D. 6

Answer 1

The Correct Option is D

We are given that the points \( P(2, 3) \), \( Q(5, k) \), and \( R(6, 7) \) are collinear. For the points to be collinear, the slope between any two points must be the same. Let's calculate the slope between points \( P \) and \( Q \), and the slope between points \( Q \) and \( R \). The slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \] 1. Slope between \( P(2, 3) \) and \( Q(5, k) \): \[ \text{slope of PQ} = \frac{k - 3}{5 - 2} = \frac{k - 3}{3} \] 2. Slope between \( Q(5, k) \) and \( R(6, 7) \): \[ \text{slope of QR} = \frac{7 - k}{6 - 5} = 7 - k \] Since the points are collinear, the slopes must be equal: \[ \frac{k - 3}{3} = 7 - k \] Now, solve for \( k \): \[ k - 3 = 3(7 - k) \] \[ k - 3 = 21 - 3k \] \[ k + 3k = 21 + 3 \] \[ 4k = 24 \] \[ k = 6 \]

The correct option is (D): \(6\)