A rectangular hyperbola of latus rectum 4 units passes through (0,0) and has (2,0) as its one focus. The equation of locus of the other focus is

Options

A. `x^(2)+y^(2) =36`
B. `x^(2)+y^(2) =4`
C. `x^(2)-y^(2) =4`
D. `x^(2)+y^(2)=9`

Explanation

To solve the problem, we need to find the equation of the locus of the other focus of a rectangular hyperbola given certain conditions. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Properties of the Hyperbola**:
- A rectangular hyperbola has the property that the latus rectum is equal to the transverse axis.
- The latus rectum (L) is given as 4 units.

2. **Identifying the Foci**:
- One focus is given as \( S = (2, 0) \).
- Let the other focus be \( S' = (h, k) \).

3. **Using the Definition of a Hyperbola**:
- For any point \( P(x, y) \) on the hyperbola, the difference of the distances from \( P \) to the foci is constant and equal to the length of the transverse axis (which is also equal to the latus rectum in a rectangular hyperbola).
- The distance between the foci is given by \( 2c \), where \( c \) is the distance from the center to each focus.

4. **Finding the Distance**:
- The distance from \( P(0, 0) \) to \( S(2, 0) \) is \( SP = \sqrt{(0 - 2)^2 + (0 - 0)^2} = 2 \).
- The distance from \( P(0, 0) \) to \( S'(h, k) \) is \( S'P = \sqrt{(0 - h)^2 + (0 - k)^2} = \sqrt{h^2 + k^2} \).

5. **Setting Up the Equation**:
- According to the property of hyperbolas, we have:
\[ |S'P - SP| = 4 \]
- This leads to two cases:
- Case 1: \( S'P - SP = 4 \)
- Case 2: \( SP - S'P = 4 \)

6. **Solving Case 1**:
- From Case 1:
\[ \sqrt{h^2 + k^2} - 2 = 4 \]
\[ \sqrt{h^2 + k^2} = 6 \]
- Squaring both sides:
\[ h^2 + k^2 = 36 \]

7. **Conclusion**:
- The equation \( h^2 + k^2 = 36 \) represents a circle with a radius of 6 centered at the origin.
- Replacing \( h \) and \( k \) with \( x \) and \( y \), we get:
\[ x^2 + y^2 = 36 \]

### Final Answer:
The equation of the locus of the other focus is:
\[ x^2 + y^2 = 36 \]