Focus of parabola y = `ax^(2) + bx + c` is

Options

A. `((-b)/(2a).(b^(2)-4ac+1)/(4a))`
B. `((-b)/(2a).(4ac-b^(2)+1)/(4a))`
C. `((-b)/(2a).(b^(2)-4ac-1)/(4a))`
D. `((b)/(2a).(b^(2)-4ac+1)/(4a))`

Answer

To find the focus of the parabola given by the equation \( y = ax^2 + bx + c \), we can follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation:
\[ y = ax^2 + bx + c \]
To rearrange this into a standard form, we divide through by \( a \):
\[ \frac{y}{a} = x^2 + \frac{b}{a}x + \frac{c}{a} \]

### Step 2: Complete the square
Next, we complete the square for the quadratic expression in \( x \):
\[ x^2 + \frac{b}{a}x = \left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 \]
Substituting this back into our equation gives:
\[ \frac{y}{a} = \left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2} + \frac{c}{a} \]
Rearranging this, we have:
\[ \frac{y}{a} + \frac{b^2}{4a^2} - \frac{c}{a} = \left(x + \frac{b}{2a}\right)^2 \]

### Step 3: Isolate the squared term
Now we isolate the squared term:
\[ \left(x + \frac{b}{2a}\right)^2 = \frac{y}{a} + \frac{b^2}{4a^2} - \frac{c}{a} \]

### Step 4: Express in standard parabola form
We can express this in the form:
\[ \left(x - h\right)^2 = 4p\left(y - k\right) \]
where \( h = -\frac{b}{2a} \) and \( k = \frac{4ac - b^2}{4a} \).

### Step 5: Identify \( p \)
From our rearranged equation, we can identify \( p \):
\[ 4p = \frac{1}{a} \implies p = \frac{1}{4a} \]

### Step 6: Find the focus
The focus of the parabola is given by the coordinates:
\[ (h, k + p) \]
Substituting our values for \( h \), \( k \), and \( p \):
\[ \text{Focus} = \left(-\frac{b}{2a}, \frac{4ac - b^2}{4a} + \frac{1}{4a}\right) \]
This simplifies to:
\[ \text{Focus} = \left(-\frac{b}{2a}, \frac{4ac - b^2 + 1}{4a}\right) \]

### Final Result
Thus, the focus of the parabola \( y = ax^2 + bx + c \) is:
\[ \left(-\frac{b}{2a}, \frac{4ac - b^2 + 1}{4a}\right) \]

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