Suppose f(x,y)=0 is the equation of a circle such that f(x,1)=0 has equal roots (each equal to 2) and f(1,x)=0 also has equal roots (each equal to zero). The equation of circle is

Options

A. `x^(2)+y^(2)+4x+3=0`
B. `x^(2)+y^(2)+4y+3=0`
C. `x^(2)+y^(2)+4x-3=0`
D. `x^(2)+y^(2)-4x+3=0`

Answer

To solve the problem, we need to find the equation of a circle given the conditions on its roots. The general form of the equation of a circle is:

\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

### Step 1: Analyze the first condition \( f(x, 1) = 0 \)

From the problem, we know that \( f(x, 1) = 0 \) has equal roots, each equal to 2. This means we can substitute \( y = 1 \) into the general equation:

\[ f(x, 1) = x^2 + 1^2 + 2gx + 2f(1) + c = 0 \]

This simplifies to:

\[ x^2 + 1 + 2gx + 2f + c = 0 \]

Since the roots are equal and both equal to 2, we can express this quadratic equation in the form:

\[ (x - 2)^2 = 0 \]

Expanding this gives:

\[ x^2 - 4x + 4 = 0 \]

### Step 2: Compare coefficients

Now, we can compare coefficients from the two equations:

1. From \( x^2 + 1 + 2gx + 2f + c = 0 \):
- Coefficient of \( x \): \( 2g = -4 \) ⇒ \( g = -2 \)
- Constant term: \( 1 + 2f + c = 4 \) ⇒ \( 2f + c = 3 \) (Equation 1)

### Step 3: Analyze the second condition \( f(1, x) = 0 \)

Next, we analyze the second condition \( f(1, x) = 0 \), which also has equal roots, each equal to 0. Substituting \( x = 1 \) into the general equation gives:

\[ f(1, x) = 1^2 + x^2 + 2g(1) + 2fx + c = 0 \]

This simplifies to:

\[ 1 + x^2 + 2g + 2fx + c = 0 \]

Since the roots are equal and both equal to 0, we can express this quadratic equation as:

\[ (x - 0)^2 = 0 \]

Expanding gives:

\[ x^2 = 0 \]

### Step 4: Compare coefficients again

Now, we can compare coefficients from the two equations:

1. From \( 1 + x^2 + 2g + 2fx + c = 0 \):
- Coefficient of \( x \): \( 2f = 0 \) ⇒ \( f = 0 \)
- Constant term: \( 1 + 2g + c = 0 \) ⇒ \( 2g + c = -1 \) (Equation 2)

### Step 5: Solve the equations

Now we have two equations:
1. \( 2f + c = 3 \) (Equation 1)
2. \( 2g + c = -1 \) (Equation 2)

Substituting \( f = 0 \) into Equation 1 gives:

\[ c = 3 \]

Substituting \( g = -2 \) into Equation 2 gives:

\[ 2(-2) + c = -1 \implies -4 + c = -1 \implies c = 3 \]

### Step 6: Write the equation of the circle

Now we have \( g = -2 \), \( f = 0 \), and \( c = 3 \). Substituting these values back into the general form of the circle gives:

\[ x^2 + y^2 + 2(-2)x + 2(0)y + 3 = 0 \]

This simplifies to:

\[ x^2 + y^2 - 4x + 3 = 0 \]

Thus, the equation of the circle is:

\[ \boxed{x^2 + y^2 - 4x + 3 = 0} \]