The shortest distance between the two lines `L_1:x=k_1, y=k_2 and L_2: x=k_3, y=k_4` is equal to

Options

A. `|sqrt(k_1^2+k_2^2)-sqrt(k_3^2+k_4^2)|`
B. `sqrt(k_1k_3+k_3k_4)`
C. `sqrt((k_1+k_3)^2+(k_2+k_4)^2)`
D. `sqrt((k_1-k_3)^2+(k_2-k_4)^2)`

Answer

To find the shortest distance between the two lines \( L_1: x = k_1, y = k_2 \) and \( L_2: x = k_3, y = k_4 \), we can follow these steps:

### Step 1: Represent the Lines in Vector Form
The first line \( L_1 \) can be represented in vector form as:
\[ \mathbf{A_1} = k_1 \hat{i} + k_2 \hat{j} + 0 \hat{k} \]
This indicates that \( L_1 \) is parallel to the z-axis.

The second line \( L_2 \) can be represented as:
\[ \mathbf{A_2} = k_3 \hat{i} + k_4 \hat{j} + 0 \hat{k} \]
This also indicates that \( L_2 \) is parallel to the z-axis.

### Step 2: Find the Direction Vectors
The direction vector \( \mathbf{B} \) for both lines is:
\[ \mathbf{B} = 0 \hat{i} + 0 \hat{j} + 1 \hat{k} \]
This shows that both lines are vertical and parallel to the z-axis.

### Step 3: Calculate the Vector Between the Two Points
The vector \( \mathbf{A_2} - \mathbf{A_1} \) is given by:
\[ \mathbf{A_2} - \mathbf{A_1} = (k_3 - k_1) \hat{i} + (k_4 - k_2) \hat{j} + 0 \hat{k} \]

### Step 4: Compute the Cross Product
To find the shortest distance, we need to compute the cross product \( \mathbf{B} \times (\mathbf{A_2} - \mathbf{A_1}) \):
\[ \mathbf{B} \times (\mathbf{A_2} - \mathbf{A_1}) = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 0 & 1 \\ k_3 - k_1 & k_4 - k_2 & 0 \end{vmatrix} \]
Calculating this determinant, we get:
\[ = \hat{i}(0) - \hat{j}(0) + \hat{k}(0 - (k_4 - k_2)) = (k_2 - k_4) \hat{i} + (k_1 - k_3) \hat{j} \]

### Step 5: Find the Magnitude of the Cross Product
The magnitude of the cross product is:
\[ |\mathbf{B} \times (\mathbf{A_2} - \mathbf{A_1})| = \sqrt{(k_2 - k_4)^2 + (k_1 - k_3)^2} \]

### Step 6: Calculate the Magnitude of the Direction Vector
The magnitude of the direction vector \( \mathbf{B} \) is:
\[ |\mathbf{B}| = \sqrt{0^2 + 0^2 + 1^2} = 1 \]

### Step 7: Use the Formula for Shortest Distance
The formula for the shortest distance \( d \) between the two lines is:
\[ d = \frac{|\mathbf{B} \times (\mathbf{A_2} - \mathbf{A_1})|}{|\mathbf{B}|} \]
Substituting the values we found:
\[ d = \frac{\sqrt{(k_2 - k_4)^2 + (k_1 - k_3)^2}}{1} = \sqrt{(k_2 - k_4)^2 + (k_1 - k_3)^2} \]

### Final Answer
Thus, the shortest distance between the two lines is:
\[ d = \sqrt{(k_2 - k_4)^2 + (k_1 - k_3)^2} \]