To solve the problem, we need to find the ordinates \( y_1, y_2, y_3 \) corresponding to the tangents to the parabola \( y^2 = 2x \) at the given angles of inclination \( 30^\circ, 40^\circ, \) and \( 50^\circ \).
### Step-by-Step Solution:
1. **Identify the Parabola**:
The given equation of the parabola is \( y^2 = 2x \). This can be rewritten in the standard form \( y^2 = 4ax \) where \( a = \frac{1}{2} \).
**Hint**: Remember that the standard form of a parabola helps identify its properties.
2. **Find the Slope of the Tangents**:
The slope \( m \) of the tangent line at any point on the parabola can be determined from the angle of inclination. The slopes corresponding to the angles \( 30^\circ, 40^\circ, \) and \( 50^\circ \) are:
\[
m_1 = \tan(30^\circ) = \frac{1}{\sqrt{3}}, \quad m_2 = \tan(40^\circ), \quad m_3 = \tan(50^\circ)
\]
**Hint**: Use the tangent function to convert angles into slopes.
3. **Equation of the Tangent**:
The equation of the tangent to the parabola \( y^2 = 2x \) at a point \( (x_0, y_0) \) is given by:
\[
yy_0 = x + \frac{y_0^2}{2}
\]
Here, \( y_0 = mx_0 \) where \( m \) is the slope of the tangent.
4. **Express \( y_0 \) in terms of \( m \)**:
Rearranging the tangent equation gives:
\[
y = mx + \frac{y_0^2}{2y_0}
\]
Since \( y_0^2 = 2x_0 \), we can substitute \( x_0 = \frac{y_0^2}{2} \) into the equation:
\[
y = mx + \frac{y_0}{2}
\]
5. **Find the ordinates \( y_1, y_2, y_3 \)**:
For each slope, we can find the corresponding ordinates:
\[
y_1 = m_1 \cdot \frac{y_1^2}{2}, \quad y_2 = m_2 \cdot \frac{y_2^2}{2}, \quad y_3 = m_3 \cdot \frac{y_3^2}{2}
\]
Rearranging gives:
\[
y_1^2 = 2y_1m_1, \quad y_2^2 = 2y_2m_2, \quad y_3^2 = 2y_3m_3
\]
6. **Solve for \( y_1, y_2, y_3 \)**:
We can find \( y_1, y_2, y_3 \) by solving the quadratic equations formed:
\[
y_1 = 2m_1, \quad y_2 = 2m_2, \quad y_3 = 2m_3
\]
7. **Determine the Ascending Order**:
Since \( m_1 < m_2 < m_3 \) (because \( 30^\circ < 40^\circ < 50^\circ \)), it follows that:
\[
y_1 < y_2 < y_3
\]
Thus, the ascending order of \( y_1, y_2, y_3 \) is:
\[
y_1, y_2, y_3
\]
### Final Answer:
The ascending order of \( y_1, y_2, y_3 \) is \( y_1, y_2, y_3 \).
