What is the value of x for which the points A (-1, 3), B (2, x) and C (5, -1) are collinear?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1
Three points are collinear (on one straight line) when the slope of AB equals the slope of AC. The slope of the line through \((x_1,y_1)\) and \((x_2,y_2)\) is \(\frac{y_2-y_1}{x_2-x_1}\).
Slope of AC \(\displaystyle =\frac{-1-3}{5-(-1)}=\frac{-4}{6}=-\frac{2}{3}\)
Slope of AB \(\displaystyle =\frac{x-3}{2-(-1)}=\frac{x-3}{3}\)
\(\displaystyle \frac{x-3}{3}=-\frac{2}{3}\), so \(x-3=-2\) and \(x=1\)
Shortcut: B’s first coordinate, 2, lies midway between −1 and 5.
On the line AC, the point with the midway \(x\) is the midpoint of AC
So its \(y\) is midway between 3 and −1: \(\frac{3+(-1)}{2}=1\)