If the focus of the parabola `(y - k)^2 = 4(x - h)` always lies between the lines `x + y = 1 and x + y = 3,` then

Options

A. `0 lt h + k lt 3`
B. `0 lt h + k lt 1`
C. `0 lt h + k lt 2`
D. `1 lt h h + k lt 3 `

Answer

To solve the problem, we need to determine the conditions under which the focus of the parabola \((y - k)^2 = 4(x - h)\) lies between the lines \(x + y = 1\) and \(x + y = 3\).

### Step-by-Step Solution:

1. **Identify the Focus of the Parabola:**
The given equation of the parabola is \((y - k)^2 = 4(x - h)\). The focus of this parabola can be identified as the point \((h + 1, k)\). This is derived from the standard form of the parabola, where the focus is located at \((h + \frac{p}{2}, k)\) with \(p = 4\).

2. **Set Up Inequalities for the Focus:**
We know that the focus \((h + 1, k)\) must lie between the two lines:
- For the line \(x + y = 1\):
\[ (h + 1) + k < 1 \]
- For the line \(x + y = 3\):
\[ (h + 1) + k > 3 \]

3. **Combine the Inequalities:**
From the inequalities derived from the lines, we can write:
\[ 1 < (h + 1) + k < 3 \]
This can be split into two separate inequalities:
- \(h + k < 0\) (from \(h + k < 1 - 1\))
- \(h + k > 2\) (from \(h + k > 3 - 1\))

4. **Final Result:**
Therefore, we have:
\[ 0 < h + k < 2 \]
This means that the sum \(h + k\) must be greater than 0 and less than 2.

### Conclusion:
The conditions for the focus of the parabola \((y - k)^2 = 4(x - h)\) to lie between the lines \(x + y = 1\) and \(x + y = 3\) can be summarized as:
\[ 0 < h + k < 2 \]