The set of real values of 'a' for which at least one tangent to `y^(2)=4ax` becomes normal to the circle
`x^(2)+y^(2)-2ax-4ay+3a^(2)-0,` is

Options

A. `[1, 2]`
B. `[sqrt2, 3]`
C. `R`
D. none of these

Explanation

To solve the problem, we need to determine the set of real values of 'a' for which at least one tangent to the parabola \( y^2 = 4ax \) becomes normal to the given circle \( x^2 + y^2 - 2ax - 4ay + 3a^2 = 0 \).

### Step-by-Step Solution:

1. **Identify the Tangent to the Parabola**:
The equation of the tangent to the parabola \( y^2 = 4ax \) can be expressed as:
\[ y = mx + \frac{a}{m} \]
where \( m \) is the slope of the tangent.

2. **Find the Center of the Circle**:
The given circle's equation is:
\[ x^2 + y^2 - 2ax - 4ay + 3a^2 = 0 \]
To find the center of the circle, we can rewrite it in standard form. The center \( (h, k) \) can be determined from the coefficients:
\[ h = \frac{2a}{2} = a, \quad k = \frac{4a}{2} = 2a \]
Thus, the center of the circle is \( (a, 2a) \).

3. **Condition for the Tangent to be Normal to the Circle**:
For the tangent line to be normal to the circle, it must pass through the center of the circle. Therefore, substituting the center \( (a, 2a) \) into the tangent line equation:
\[ 2a = ma + \frac{a}{m} \]

4. **Rearranging the Equation**:
Rearranging the equation gives:
\[ 2a = ma + \frac{a}{m} \implies 2a - ma = \frac{a}{m} \]
Dividing through by \( a \) (assuming \( a \neq 0 \)):
\[ 2 - m = \frac{1}{m} \]

5. **Multiplying through by \( m \)**:
Multiplying through by \( m \) yields:
\[ 2m - m^2 = 1 \implies m^2 - 2m + 1 = 0 \]
This simplifies to:
\[ (m - 1)^2 = 0 \]
Thus, \( m = 1 \).

6. **Conclusion about the Values of 'a'**:
Since the equation \( (m - 1)^2 = 0 \) does not depend on 'a', it implies that for every real value of \( a \), there exists at least one tangent to the parabola that is normal to the circle.

7. **Final Result**:
Therefore, the set of real values of \( a \) for which at least one tangent to the parabola becomes normal to the circle is:
\[ \text{All real numbers } a \in \mathbb{R} \]