The point on the parabola `y^(2) = 4ax` nearest to the focus has its abscissa.

Options

A. x=0
B. x=a
C. `x=(a)/(2)`
D. `x=2a`

Answer

To find the abscissa of the point on the parabola \( y^2 = 4ax \) that is nearest to the focus, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Focus of the Parabola**:
The given parabola is \( y^2 = 4ax \). The focus of this parabola is at the point \( (a, 0) \).

2. **Define a Point on the Parabola**:
Let \( P(x, y) \) be a point on the parabola. Since it lies on the parabola, it satisfies the equation \( y^2 = 4ax \).

3. **Calculate the Distance from the Focus**:
The distance \( d \) from the point \( P(x, y) \) to the focus \( (a, 0) \) can be expressed using the distance formula:
\[ d = \sqrt{(x - a)^2 + (y - 0)^2} = \sqrt{(x - a)^2 + y^2} \]

4. **Substitute for \( y^2 \)**:
Since \( y^2 = 4ax \), we can substitute this into the distance formula:
\[ d = \sqrt{(x - a)^2 + 4ax} \]

5. **Simplify the Expression**:
Expanding the distance expression gives:
\[ d = \sqrt{(x - a)^2 + 4ax} = \sqrt{x^2 - 2ax + a^2 + 4ax} = \sqrt{x^2 + 2ax + a^2} \]
This can be rewritten as:
\[ d = \sqrt{(x + a)^2} \]
Therefore, we have:
\[ d = |x + a| \]

6. **Minimize the Distance**:
To find the point nearest to the focus, we need to minimize \( d \). The expression \( |x + a| \) is minimized when \( x + a = 0 \), which gives:
\[ x = -a \]

7. **Conclusion**:
The abscissa of the point on the parabola nearest to the focus is \( -a \).