Which of the following is the equation of circle

Options

A. `x^2+2y^2-x+6=0`
B. `x^2-y^2+x+y+1=0`
C. `x^2+y^2+xy+1=0`
D. `3(x^2+y^2)+5x+1=0`

Answer

To determine which of the given equations represents a circle, we need to analyze the equations based on the general form of a conic section. The general equation of a conic section can be expressed as:

\[ AX^2 + 2BXY + CY^2 + DX + EY + F = 0 \]

For an equation to represent a circle, it must satisfy the following conditions:
1. The coefficient \( B \) must be equal to 0.
2. The coefficients \( A \) and \( C \) must be equal.

Let's analyze the options step by step.

### Step 1: Identify the coefficients from each option
1. **Option 1:** \( X^2 + 2Y^2 + 3X + 4Y + 5 = 0 \)
- Coefficients: \( A = 1, B = 0, C = 2, D = 3, E = 4, F = 5 \)

2. **Option 2:** \( X^2 - Y^2 + 3X + 4Y + 5 = 0 \)
- Coefficients: \( A = 1, B = 0, C = -1, D = 3, E = 4, F = 5 \)

3. **Option 3:** \( X^2 + Y^2 + 2XY + 3X + 4Y + 5 = 0 \)
- Coefficients: \( A = 1, B = 1, C = 1, D = 3, E = 4, F = 5 \)

4. **Option 4:** \( 3X^2 + 3Y^2 + 5X + 1 = 0 \)
- Coefficients: \( A = 3, B = 0, C = 3, D = 5, E = 0, F = 1 \)

### Step 2: Check the conditions for each option
- **Option 1:**
- \( B = 0 \) (satisfied)
- \( A \neq C \) (1 ≠ 2, not satisfied)
- **Conclusion:** Not a circle.

- **Option 2:**
- \( B = 0 \) (satisfied)
- \( A \neq C \) (1 ≠ -1, not satisfied)
- **Conclusion:** Not a circle.

- **Option 3:**
- \( B \neq 0 \) (B = 1, not satisfied)
- **Conclusion:** Not a circle.

- **Option 4:**
- \( B = 0 \) (satisfied)
- \( A = C \) (3 = 3, satisfied)
- **Conclusion:** This is a circle.

### Final Answer
The equation that represents a circle is **Option 4: \( 3X^2 + 3Y^2 + 5X + 1 = 0 \)**.