To determine the values of $a$ and $b$ such that the points $A(1, 2)$, $B(4, b)$, and $C(a, 6)$ are collinear, we can use the concept of the slope of a line.

The points are collinear if the slopes between any two pairs of points are equal. We will set the slopes between points $A$ and $B$, and between points $B$ and $C$, equal to each other.

Step 1: Calculate the slope between two points.

The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Slope between $A(1, 2)$ and $B(4, b)$:

$m_{AB} = \frac{b - 2}{4 - 1} = \frac{b - 2}{3}$

Slope between $B(4, b)$ and $C(a, 6)$:

$m_{BC} = \frac{6 - b}{a - 4}$

Step 2: Set the slopes equal.

Since the points are collinear, the slopes must be equal: $\frac{b - 2}{3} = \frac{6 - b}{a - 4}$

Step 3: Solve the equation.

Cross-multiply to eliminate the denominators: $(b - 2)(a - 4) = 3(6 - b)$

Expand both sides: $b a - 4b - 2a + 8 = 18 - 3b$

Rearrange the terms: $b a - 4b + 3b - 2a = 18 - 8$ $b a - b - 2a = 10$

This is the equation we have so far. Now, we need to find $a$ and $b$.

One way to proceed is to substitute specific values for either $a$ or $b$ and solve the equation, or you can attempt to isolate one variable in terms of the other.

Would you like me to continue solving for these variables, or explain further?