Find the focus of the parabola `y^2 = 12x`

Options

A. `(0,3)`
B. `(3,0)`
C. `(0,-3)`
D. `(-3,0)`

Answer

To find the focus of the parabola given by the equation \( y^2 = 12x \), we can follow these steps:

### Step 1: Identify the standard form of the parabola
The standard form of a parabola that opens to the right is given by:
\[ y^2 = 4ax \]
where \( a \) is the distance from the vertex to the focus.

### Step 2: Compare the given equation with the standard form
We have the equation:
\[ y^2 = 12x \]
Now, we can compare this with the standard form \( y^2 = 4ax \).

### Step 3: Determine the value of \( a \)
From the comparison, we see that:
\[ 4a = 12 \]
To find \( a \), we divide both sides by 4:
\[ a = \frac{12}{4} = 3 \]

### Step 4: Find the coordinates of the focus
For a parabola of the form \( y^2 = 4ax \), the focus is located at the point \( (a, 0) \). Since we found \( a = 3 \), the coordinates of the focus are:
\[ (3, 0) \]

### Conclusion
Thus, the focus of the parabola \( y^2 = 12x \) is at the point:
\[ \boxed{(3, 0)} \]

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