To solve the problem of matching the items in Column-I with those in Column-II, we will analyze each item in Column-I step by step.
### Step 1: Analyze Column (A)
We have the equation:
\[
|z - 6i| + |z - 8| = k
\]
This represents an ellipse if \( k \) is greater than the distance between the two points \( 6i \) and \( 8 \).
1. **Identify the points**: The points are \( (0, 6) \) and \( (8, 0) \).
2. **Calculate the distance**:
\[
\text{Distance} = \sqrt{(8 - 0)^2 + (0 - 6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\]
3. **Condition for ellipse**: For the equation to represent an ellipse, \( k \) must be greater than 10.
4. **Conclusion**: The value of \( k \) must not equal 10, so \( k \) can be \( p, q, r, s \) (2, 8, 12, 11).
### Step 2: Analyze Column (B)
We have the equation:
\[
||z - 12i + 3| - |z - 2|| = k
\]
This represents a hyperbola if \( k \) is less than the distance between the two points \( (0, 12) \) and \( (2, 0) \).
1. **Identify the points**: The points are \( (0, 12) \) and \( (2, 0) \).
2. **Calculate the distance**:
\[
\text{Distance} = \sqrt{(2 - 0)^2 + (0 - 12)^2} = \sqrt{4 + 144} = \sqrt{148} = \sqrt{4 \times 37} = 2\sqrt{37}
\]
Approximating \( \sqrt{37} \approx 6.08 \), we have \( 2\sqrt{37} \approx 12.16 \).
3. **Condition for hyperbola**: For the equation to represent a hyperbola, \( k \) must not equal the distance, which is approximately 12.16.
4. **Conclusion**: Since there is no option for \( k = 12.16 \), \( k \) can be \( p, q, r, s, t \) (2, 8, 12, 11, 10).
### Step 3: Analyze Column (C)
We have the equation:
\[
|z - ki| + |z - 4| = \sqrt{10k}
\]
This represents a line segment if the distance between the two points equals \( \sqrt{10k} \).
1. **Identify the points**: The points are \( (0, k) \) and \( (4, 0) \).
2. **Calculate the distance**:
\[
\text{Distance} = \sqrt{(4 - 0)^2 + (0 - k)^2} = \sqrt{16 + k^2}
\]
3. **Condition for line segment**: For the equation to represent a line segment, we need:
\[
\sqrt{16 + k^2} = \sqrt{10k}
\]
Squaring both sides:
\[
16 + k^2 = 10k \implies k^2 - 10k + 16 = 0
\]
Using the quadratic formula:
\[
k = \frac{10 \pm \sqrt{(10)^2 - 4 \cdot 1 \cdot 16}}{2 \cdot 1} = \frac{10 \pm \sqrt{100 - 64}}{2} = \frac{10 \pm \sqrt{36}}{2} = \frac{10 \pm 6}{2}
\]
This gives us \( k = 8 \) and \( k = 2 \).
4. **Conclusion**: The values of \( k \) are \( 2 \) and \( 8 \), so the answer is \( p \) and \( q \).
### Step 4: Analyze Column (D)
We have the equation:
\[
\frac{z - k + 2ki}{|z - 2 + 4i|} = k
\]
This represents a circle if \( k \) is not equal to 1.
1. **Rearranging**: We can express this in terms of modulus:
\[
|z - k + 2ki| = k |z - 2 + 4i|
\]
2. **Condition for circle**: The equation represents a circle if the constant \( k \) is not equal to 1.
3. **Conclusion**: Since \( k \) can be \( p, q, r, s, t \) (2, 8, 12, 11, 10), all options are valid.
### Final Matching
- (A) matches with \( (p) 2 \)
- (B) matches with \( (q) 8 \)
- (C) matches with \( (p) 2 \) and \( (q) 8 \)
- (D) matches with \( (p, q, r, s, t) \)
### Summary of Matches
- A → (p) 2
- B → (q) 8
- C → (p) 2, (q) 8
- D → (p, q, r, s, t)
