The focal distance of a point on the parabola `y^2 = 12 x` is 4. What is the abscissa of the point?

Options

A. 1
B. `-1`
C. `2sqrt(3)`
D. `-2`

Answer

To solve the problem, we need to find the abscissa (x-coordinate) of a point on the parabola given by the equation \( y^2 = 12x \), where the focal distance is 4.

### Step-by-step Solution:

1. **Identify the standard form of the parabola**:
The given equation of the parabola is \( y^2 = 12x \). This can be rewritten in the standard form \( y^2 = 4ax \). Here, we can identify \( 4a = 12 \).

2. **Find the value of 'a'**:
From \( 4a = 12 \), we can solve for \( a \):
\[ a = \frac{12}{4} = 3 \]

3. **Use the formula for focal distance**:
The focal distance \( d \) from a point on the parabola to the focus is given by the formula:
\[ d = a + x \]
where \( d \) is the focal distance, \( a \) is the value we found, and \( x \) is the abscissa we want to find.

4. **Set up the equation with the given focal distance**:
We are given that the focal distance \( d = 4 \). Plugging in the values we have:
\[ 4 = a + x \]
Substituting \( a = 3 \):
\[ 4 = 3 + x \]

5. **Solve for x**:
Rearranging the equation to find \( x \):
\[ x = 4 - 3 = 1 \]

6. **Conclusion**:
The abscissa of the point on the parabola is \( x = 1 \).

### Final Answer:
The abscissa of the point is \( \boxed{1} \).

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