Question

If the points $(1, 4)$, $(a, -2)$, and $(-3, 16)$ are collinear, then find the value of $a$.

Hint:

For three points to be collinear, their slopes must be equal. Equate the slopes and solve for the variable.

Answer 1

Step 1: Recall the condition for collinearity.
Three points are collinear if the slopes between them are equal. \[ \text{Slope of } AB = \text{Slope of } BC \] Step 2: Compute slopes.
Let $A(1, 4)$, $B(a, -2)$, and $C(-3, 16)$. \[ \text{Slope of } AB = \dfrac{-2 - 4}{a - 1} = \dfrac{-6}{a - 1} \] \[ \text{Slope of } BC = \dfrac{16 - (-2)}{-3 - a} = \dfrac{18}{-3 - a} = \dfrac{-18}{a + 3} \] Step 3: Equate the slopes.
\[ \dfrac{-6}{a - 1} = \dfrac{-18}{a + 3} \Rightarrow 6(a + 3) = 18(a - 1) \] \[ 6a + 18 = 18a - 18 \Rightarrow 12a = 36 \Rightarrow a = 3 \] Step 4: Conclusion.
Hence, $a = 3$.