If two tangents can be drawn the different branches of hyperbola `(x^(2))/(1)-(y^(2))/(4) =1` from `(alpha, alpha^(2))`, then

Options

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

Explanation

To solve the problem, we need to determine the range of \(\alpha\) such that two tangents can be drawn from the point \((\alpha, \alpha^2)\) to the hyperbola given by the equation:

\[ \frac{x^2}{1} - \frac{y^2}{4} = 1 \]

### Step-by-Step Solution:

1. **Identify the hyperbola parameters**:
The given hyperbola can be compared to the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Here, we have:
- \(a^2 = 1 \Rightarrow a = 1\)
- \(b^2 = 4 \Rightarrow b = 2\)

2. **Find the equations of the asymptotes**:
The equations of the asymptotes for the hyperbola are given by:
\[ y = \pm \frac{b}{a} x = \pm 2x \]

3. **Determine the conditions for the point \((\alpha, \alpha^2)\)**:
For two tangents to be drawn from the point \((\alpha, \alpha^2)\) to the hyperbola, the point must lie between the two asymptotes. This means:
\[ -2\alpha < \alpha^2 < 2\alpha \]

4. **Analyze the inequality \(\alpha^2 < 2\alpha\)**:
Rearranging gives:
\[ \alpha^2 - 2\alpha < 0 \Rightarrow \alpha(\alpha - 2) < 0 \]
This inequality holds when \(\alpha\) is in the interval:
\[ 0 < \alpha < 2 \]

5. **Analyze the inequality \(-2\alpha < \alpha^2\)**:
Rearranging gives:
\[ \alpha^2 + 2\alpha > 0 \Rightarrow \alpha(\alpha + 2) > 0 \]
This inequality holds when \(\alpha < -2\) or \(\alpha > 0\).

6. **Combine the results**:
We have two conditions:
- From \(\alpha^2 < 2\alpha\): \(0 < \alpha < 2\)
- From \(-2\alpha < \alpha^2\): \(\alpha < -2\) or \(\alpha > 0\)

The valid range for \(\alpha\) that satisfies both conditions is:
\[ 0 < \alpha < 2 \]

### Final Result:
Thus, the range of \(\alpha\) for which two tangents can be drawn from the point \((\alpha, \alpha^2)\) to the hyperbola is:
\[ \alpha \in (0, 2) \]