Let S be the set of all possible values of parameter 'a' for which the points of intersection of the parabolas `y^(2)=3axandy=1/2(x^(2)+ax+5)` are concyclic. Then S contains the interval(s)

Options

A. `(-oo,-2)`
B. (-2, 0)
C. (0, 2)
D. `(2, oo)`

Answer

To solve the problem, we need to find the values of the parameter 'a' for which the points of intersection of the two given parabolas are concyclic. Here’s a step-by-step solution:

### Step 1: Write the equations of the parabolas
The equations of the parabolas given are:
1. \( y^2 = 3ax \)
2. \( y = \frac{1}{2}(x^2 + ax + 5) \)

### Step 2: Rearrange the second equation
To make it easier to work with, we can rearrange the second equation:
\[ 2y = x^2 + ax + 5 \]
This can be rewritten as:
\[ x^2 + ax + 5 - 2y = 0 \]

### Step 3: Substitute \( y^2 \) into the second equation
Now, we substitute \( y^2 = 3ax \) into the equation we just rearranged. To do this, we express \( y \) in terms of \( x \) from the first equation:
\[ y = \sqrt{3ax} \]
Substituting this into the rearranged second equation gives:
\[ x^2 + ax + 5 - 2\sqrt{3ax} = 0 \]

### Step 4: Form a new equation
Now we have a quadratic in \( x \):
\[ x^2 + ax + (5 - 2\sqrt{3ax}) = 0 \]

### Step 5: Determine the condition for concyclicity
For the points of intersection to be concyclic, the discriminant of this quadratic must be non-negative. The discriminant \( D \) of a quadratic \( Ax^2 + Bx + C = 0 \) is given by:
\[ D = B^2 - 4AC \]
In our case:
- \( A = 1 \)
- \( B = a \)
- \( C = 5 - 2\sqrt{3ax} \)

Thus, the discriminant becomes:
\[ D = a^2 - 4(1)(5 - 2\sqrt{3ax}) = a^2 - 20 + 8\sqrt{3ax} \]

### Step 6: Set the discriminant greater than or equal to zero
For the points to be concyclic, we need:
\[ a^2 - 20 + 8\sqrt{3ax} \geq 0 \]

### Step 7: Analyze the condition
To ensure that this inequality holds for all points of intersection, we need to analyze the term \( 8\sqrt{3ax} \). This term must be able to compensate for \( a^2 - 20 \).

### Step 8: Find the critical points
To find the values of \( a \) that satisfy this condition, we can analyze the equation:
\[ a^2 - 20 + 8\sqrt{3ax} \geq 0 \]
This leads us to:
\[ a^2 - 20 > 0 \implies a^2 > 20 \implies |a| > \sqrt{20} \implies |a| > 2\sqrt{5} \]

### Step 9: Determine the intervals
This implies:
\[ a < -2\sqrt{5} \quad \text{or} \quad a > 2\sqrt{5} \]
Thus, the set \( S \) of all possible values of \( a \) is:
\[ S = (-\infty, -2\sqrt{5}) \cup (2\sqrt{5}, \infty) \]

### Final Answer
The intervals for the parameter 'a' are:
\[ S = (-\infty, -2\sqrt{5}) \cup (2\sqrt{5}, \infty) \]