The circle `x^(2) + y^(2) + 2g x + 2fy + c = 0 ` does not intersect the y-axis if

Options

A. A. `g^(2) lt c`
B. B. ` f^(2) lt c `
C. C. ` 4f^(2) lt c`
D. D. ` f^(2) lt 4c`

Answer

To determine the condition under which the circle given by the equation \( x^2 + y^2 + 2gx + 2fy + c = 0 \) does not intersect the y-axis, we can follow these steps:

### Step 1: Substitute \( x = 0 \)
To find the intersection of the circle with the y-axis, we set \( x = 0 \) in the circle's equation. This gives us:
\[ 0^2 + y^2 + 2g(0) + 2fy + c = 0 \]
which simplifies to:
\[ y^2 + 2fy + c = 0 \]

### Step 2: Identify the coefficients
The equation \( y^2 + 2fy + c = 0 \) is a quadratic equation in \( y \). Here, the coefficients are:
- \( a = 1 \)
- \( b = 2f \)
- \( c = c \)

### Step 3: Calculate the discriminant
The condition for a quadratic equation \( ay^2 + by + c = 0 \) to have no real roots (which means it does not intersect the y-axis) is that the discriminant \( D \) must be less than 0. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Substituting our coefficients, we get:
\[ D = (2f)^2 - 4(1)(c) = 4f^2 - 4c \]

### Step 4: Set the discriminant less than zero
For the circle not to intersect the y-axis, we need:
\[ 4f^2 - 4c < 0 \]

### Step 5: Simplify the inequality
Dividing the entire inequality by 4, we have:
\[ f^2 - c < 0 \]
which can be rearranged to:
\[ f^2 < c \]

### Conclusion
Thus, the condition for the circle not to intersect the y-axis is:
\[ f^2 < c \]

### Final Answer
The correct option is **(b) \( f^2 < c \)**.