`ax ^(2) + 2y ^(2) + 2bx y +2x -y + x=0` represents a circle through the origin, if

Options

A. `a=0, b=0, c=2`
B. `a =1, b =0 , c=0`
C. `a=2, b =2, c =0`
D. `a =2, b =0 , c=0`

Answer

To determine the conditions under which the equation \( ax^2 + 2y^2 + 2bxy + 2x - y + c = 0 \) represents a circle that passes through the origin, we can follow these steps:

### Step 1: Identify the General Form of a Circle
The general equation of a circle can be expressed as:
\[ Ax^2 + Bxy + Cy^2 + 2Dx + 2Ey + F = 0 \]
For the equation to represent a circle, the following conditions must be satisfied:
1. \( A = C \) (the coefficients of \( x^2 \) and \( y^2 \) must be equal)
2. \( B = 0 \) (the coefficient of \( xy \) must be zero)
3. \( D^2 + E^2 - AF = 0 \) (this is the condition for the circle)

### Step 2: Compare Coefficients
From the given equation \( ax^2 + 2y^2 + 2bxy + 2x - y + c = 0 \), we can identify:
- \( A = a \)
- \( B = 2b \)
- \( C = 2 \)
- \( D = 1 \)
- \( E = -\frac{1}{2} \)
- \( F = c \)

### Step 3: Apply the Conditions
1. **Condition 1**: \( A = C \)
\[ a = 2 \]

2. **Condition 2**: \( B = 0 \)
\[ 2b = 0 \implies b = 0 \]

3. **Condition 3**: \( D^2 + E^2 - AF = 0 \)
\[ 1^2 + \left(-\frac{1}{2}\right)^2 - (2)(c) = 0 \]
\[ 1 + \frac{1}{4} - 2c = 0 \]
\[ \frac{5}{4} - 2c = 0 \implies 2c = \frac{5}{4} \implies c = \frac{5}{8} \]

### Step 4: Conclusion
Thus, for the equation \( ax^2 + 2y^2 + 2bxy + 2x - y + c = 0 \) to represent a circle that passes through the origin, the conditions are:
- \( a = 2 \)
- \( b = 0 \)
- \( c = \frac{5}{8} \)