Prove that the equation `y^(2)+2ax+2by+c=0` represents a parabola whose axis is parallel to the axis of x. Find its vertex.

Answer

To prove that the equation \( y^2 + 2ax + 2by + c = 0 \) represents a parabola whose axis is parallel to the x-axis and to find its vertex, we will follow these steps:

### Step 1: Identify the general form of a conic section
The general form of a conic section is given by:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
For our equation \( y^2 + 2ax + 2by + c = 0 \), we can identify the coefficients:
- \( A = 0 \)
- \( B = 0 \)
- \( C = 1 \)
- \( D = 2a \)
- \( E = 2b \)
- \( F = c \)

### Step 2: Calculate the discriminant
The discriminant \( \Delta \) for conic sections is given by:
\[ \Delta = B^2 - 4AC \]
Substituting the values we identified:
\[ \Delta = 0^2 - 4(0)(1) = 0 \]
Since \( \Delta = 0 \), this indicates that the equation represents a parabola.

### Step 3: Check the condition for a parabola
For the equation to represent a parabola, we also check the condition:
\[ h^2 - ab = 0 \]
Where \( h = \frac{D}{2} \) and \( a = A \), \( b = C \). Here:
- \( h = \frac{2a}{2} = a \)
- \( a = 0 \)
- \( b = 1 \)

Calculating \( h^2 - ab \):
\[ h^2 - ab = a^2 - (0)(1) = a^2 \]
Since \( a^2 = 0 \), this condition is satisfied.

### Step 4: Rearranging the equation
Now, we rearrange the original equation to find the vertex:
\[ y^2 + 2by = -2ax - c \]
Completing the square for the \( y \) terms:
\[ y^2 + 2by + b^2 = -2ax - c + b^2 \]
This can be rewritten as:
\[ (y + b)^2 = -2ax - c + b^2 \]
Thus, we have:
\[ (y + b)^2 = -2a\left(x + \frac{c - b^2}{2a}\right) \]

### Step 5: Identify the vertex
From the equation \( (y + b)^2 = -2a\left(x + \frac{c - b^2}{2a}\right) \), we can identify the vertex of the parabola:
- The vertex \( (h, k) \) is given by:
\[ \left(-\frac{c - b^2}{2a}, -b\right) \]

### Conclusion
Thus, we have proved that the equation \( y^2 + 2ax + 2by + c = 0 \) represents a parabola whose axis is parallel to the x-axis, and the vertex is:
\[ \left(-\frac{c - b^2}{2a}, -b\right) \]