To find the value of \( k \) for which the points \( A(0, 1) \), \( B(2, k) \), and \( C(4, -5) \) are collinear, we can use the area of the triangle formed by these points. If the area of the triangle is zero, then the points are collinear.
The formula for the area of a triangle given its vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
### Step 1: Substitute the coordinates into the area formula
Let \( A(0, 1) \), \( B(2, k) \), and \( C(4, -5) \). We substitute these coordinates into the area formula:
\[
\text{Area} = \frac{1}{2} \left| 0(k - (-5)) + 2(-5 - 1) + 4(1 - k) \right|
\]
### Step 2: Simplify the expression
Calculating the terms, we have:
\[
= \frac{1}{2} \left| 0 + 2(-6) + 4(1 - k) \right|
\]
\[
= \frac{1}{2} \left| -12 + 4 - 4k \right|
\]
\[
= \frac{1}{2} \left| -8 - 4k \right|
\]
### Step 3: Set the area equal to zero
For the points to be collinear, the area must be zero:
\[
\frac{1}{2} \left| -8 - 4k \right| = 0
\]
This implies:
\[
\left| -8 - 4k \right| = 0
\]
### Step 4: Solve the equation
The absolute value equation gives us:
\[
-8 - 4k = 0
\]
Solving for \( k \):
\[
-4k = 8
\]
\[
k = -2
\]
### Final Answer
Thus, the value of \( k \) for which the points \( A(0, 1) \), \( B(2, k) \), and \( C(4, -5) \) are collinear is:
\[
\boxed{-2}
\]
