The set of real value of 'a' for which at least one tangent to the parabola `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. `phi` (no set)

Answer

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

### Step-by-Step Solution:

1. **Identify the Circle's Center and Radius:**
The equation of the circle is given as:
\[ x^2 + y^2 - 2ax - 4ay + 3a^2 = 0 \]
We can rewrite this in standard form by completing the square:
\[ (x - a)^2 + (y - 2a)^2 = a^2 - 3a^2 = -2a^2 \]
This indicates that the center of the circle is \( (a, 2a) \) and the radius is \( \sqrt{2a^2} = \sqrt{2}|a| \).

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

3. **Normal to the Circle:**
The normal to the circle at any point passes through the center of the circle. Therefore, we need to find the conditions under which the tangent to the parabola is normal to the circle.

4. **Condition for Tangent to be Normal:**
For the tangent to be normal to the circle, the slope of the tangent line \( m \) must equal the slope of the radius at the point of tangency. The slope of the radius from the center \( (a, 2a) \) to the point on the tangent line must be perpendicular to the slope of the tangent line.

The slope of the radius from the center \( (a, 2a) \) to a point on the tangent line can be derived from the tangent equation. Setting the condition for perpendicularity gives us:
\[ m \cdot \left(-\frac{1}{m}\right) = -1 \]
This simplifies to:
\[ m^2 = 1 \implies m = 1 \text{ or } m = -1 \]

5. **Finding Values of 'a':**
Since the slope \( m \) does not depend on \( a \), we conclude that for any real value of \( a \), there exists a tangent to the parabola that can be normal to the circle. Therefore, the set of real values of \( a \) is:
\[ a \in \mathbb{R} \]

### Conclusion:
The set of real values of \( a \) for which at least one tangent to the parabola becomes normal to the circle is:
\[ \text{All real numbers } a \in \mathbb{R} \]