To solve the problem of finding the values of \( \alpha \) from which two tangents can be drawn to the different branches of the hyperbola given by the equation
\[
\frac{x^2}{1} - \frac{y^2}{4} = 1
\]
from the point \( (\alpha, \alpha^2) \), we can follow these steps:
### Step 1: Identify the Hyperbola Parameters
The hyperbola can be rewritten in the standard form:
\[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
\]
where \( a^2 = 1 \) and \( b^2 = 4 \). Thus, we have \( a = 1 \) and \( b = 2 \).
### Step 2: Write the Equation of Tangents
The equations of the tangents to the hyperbola from a point \( (x_0, y_0) \) can be given by:
\[
\frac{xx_0}{a^2} - \frac{yy_0}{b^2} = 1
\]
Substituting \( a = 1 \), \( b = 2 \), and \( (x_0, y_0) = (\alpha, \alpha^2) \):
\[
\frac{x\alpha}{1} - \frac{y\alpha^2}{4} = 1
\]
This simplifies to:
\[
\alpha x - \frac{\alpha^2 y}{4} = 1
\]
### Step 3: Determine Conditions for Two Tangents
For two tangents to exist, the point \( (\alpha, \alpha^2) \) must lie between the two asymptotes of the hyperbola. The equations of the asymptotes are:
\[
y = \pm \frac{b}{a} x = \pm 2x
\]
Thus, the point \( (\alpha, \alpha^2) \) must satisfy:
\[
-2\alpha < \alpha^2 < 2\alpha
\]
### Step 4: Solve the Inequalities
1. **For the left inequality**:
\[
\alpha^2 > -2\alpha
\]
Rearranging gives:
\[
\alpha^2 + 2\alpha > 0 \implies \alpha(\alpha + 2) > 0
\]
The critical points are \( \alpha = 0 \) and \( \alpha = -2 \). The sign chart gives us:
- \( \alpha < -2 \): Positive
- \( -2 < \alpha < 0 \): Negative
- \( \alpha > 0 \): Positive
Thus, \( \alpha \in (-\infty, -2) \cup (0, \infty) \).
2. **For the right inequality**:
\[
\alpha^2 < 2\alpha
\]
Rearranging gives:
\[
\alpha^2 - 2\alpha < 0 \implies \alpha(\alpha - 2) < 0
\]
The critical points are \( \alpha = 0 \) and \( \alpha = 2 \). The sign chart gives us:
- \( \alpha < 0 \): Positive
- \( 0 < \alpha < 2 \): Negative
- \( \alpha > 2 \): Positive
Thus, \( \alpha \in (0, 2) \).
### Step 5: Find the Intersection of the Two Regions
Now we find the intersection of the two sets:
1. From the first inequality: \( (-\infty, -2) \cup (0, \infty) \)
2. From the second inequality: \( (0, 2) \)
The intersection is \( (0, 2) \).
### Conclusion
Thus, the values of \( \alpha \) from which two tangents can be drawn to the hyperbola from the point \( (\alpha, \alpha^2) \) are:
\[
\alpha \in (0, 2)
\]
