Find the distance between the points `(at_(1)^(2), 2 at_(1))` and `(at_(2)^(2), 2 at_(2))`, where `t_(1)` and `t_(2)` are the roots of the equation `x^(2)-2sqrt(3)x+2=0` and `a gt 0`.

Answer

To find the distance between the points \((at_1^2, 2at_1)\) and \((at_2^2, 2at_2)\), where \(t_1\) and \(t_2\) are the roots of the equation \(x^2 - 2\sqrt{3}x + 2 = 0\), we can follow these steps:

### Step 1: Find the roots \(t_1\) and \(t_2\)
The roots of the quadratic equation \(x^2 - 2\sqrt{3}x + 2 = 0\) can be found using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -2\sqrt{3}\), and \(c = 2\).

Calculating the discriminant:
\[ b^2 - 4ac = (2\sqrt{3})^2 - 4 \cdot 1 \cdot 2 = 12 - 8 = 4 \]
Now substituting into the quadratic formula:
\[ t_{1,2} = \frac{2\sqrt{3} \pm \sqrt{4}}{2} = \frac{2\sqrt{3} \pm 2}{2} = \sqrt{3} \pm 1 \]
Thus, the roots are:
\[ t_1 = \sqrt{3} + 1, \quad t_2 = \sqrt{3} - 1 \]

### Step 2: Calculate the distance between the points
The points are:
\[ A = (at_1^2, 2at_1) = (a(\sqrt{3}+1)^2, 2a(\sqrt{3}+1)) \]
\[ B = (at_2^2, 2at_2) = (a(\sqrt{3}-1)^2, 2a(\sqrt{3}-1)) \]

Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \((x_1, y_1) = (at_1^2, 2at_1)\) and \((x_2, y_2) = (at_2^2, 2at_2)\).

Substituting the coordinates:
\[ d = \sqrt{(a(\sqrt{3}-1)^2 - a(\sqrt{3}+1)^2)^2 + (2a(\sqrt{3}-1) - 2a(\sqrt{3}+1))^2} \]

### Step 3: Simplify the expressions
1. **For the x-coordinates:**
\[ a((\sqrt{3}-1)^2 - (\sqrt{3}+1)^2) = a((3 - 2\sqrt{3} + 1) - (3 + 2\sqrt{3} + 1)) = a(4 - 2\sqrt{3} - 4 - 2\sqrt{3}) = -4a\sqrt{3} \]

2. **For the y-coordinates:**
\[ 2a((\sqrt{3}-1) - (\sqrt{3}+1)) = 2a(-2) = -4a \]

### Step 4: Substitute back into the distance formula
\[ d = \sqrt{(-4a\sqrt{3})^2 + (-4a)^2} \]
Calculating:
\[ = \sqrt{16a^2 \cdot 3 + 16a^2} = \sqrt{48a^2 + 16a^2} = \sqrt{64a^2} = 8a \]

### Final Answer
The distance between the points is:
\[ \boxed{8a} \]