To solve this problem, let's break it down step by step, using the information visible in the figure.

1. General Observations:

• The figure appears to show a parabola with a focus $F$ and directrix below it.
• The coordinates of point $A$ are directly above $F$, located vertically on the parabola.
• $B$ is a point on the parabola, to the right of $F$, and connected via the line $OB$ which forms an angle $\alpha$ with the horizontal axis.

The goal is to express:

• Coordinates of $A$ and $B$,
• Distances $|AB|$ and $|OB|$, in terms of $h$ (the height of $A$ above $F$) and $\alpha$ (the angle of $OB$).

2. Coordinate Geometry of the Parabola:

A parabola with a focus $F(0, f)$ and a directrix $y = d$ is described by: $(x - 0)^2 = 4p(y - f),$ where $p = f - d$ is the focal length.

Specifics for the problem:

• The directrix appears to lie on the $x$-axis, meaning $d = 0$. Hence, $f = p$.
• The parabola's equation simplifies to: $x^2 = 4p(y - p),$ where $F(0, p)$.

3. Coordinates of Point $A$:

• $A$ lies directly above $F$, at a vertical distance $h$ from $F(0, p)$. Therefore: $A = (0, p + h).$

4. Coordinates of Point $B$:

• $B$ lies on the parabola and is determined by the angle $\alpha$ and the equation of the parabola.
• Let $B(x_B, y_B)$. From the parabola's equation: $x_B^2 = 4p(y_B - p).$
• Since $OB$ makes an angle $\alpha$ with the $x$-axis: $\tan(\alpha) = \frac{y_B}{x_B} \quad \implies \quad y_B = x_B \tan(\alpha).$
• Substituting $y_B = x_B \tan(\alpha)$ into the parabola's equation: $x_B^2 = 4p(x_B \tan(\alpha) - p).$
• Rearranging: $x_B^2 = 4p x_B \tan(\alpha) - 4p^2.$
• Solving for $x_B$: $x_B (x_B - 4p \tan(\alpha)) = -4p^2.$ From here, $x_B$ can be solved explicitly in terms of $p$ and $\tan(\alpha)$. After solving for $x_B$, substitute back to find $y_B$.

5. Distance $|AB|$:

The distance between $A(0, p + h)$ and $B(x_B, y_B)$ is given by: $|AB| = \sqrt{(x_B - 0)^2 + (y_B - (p + h))^2}.$

6. Distance $|OB|$:

The distance between $O(0, 0)$ and $B(x_B, y_B)$ is: $|OB| = \sqrt{x_B^2 + y_B^2}.$ Substitute $y_B = x_B \tan(\alpha)$ into this expression.

Next Steps:

Would you like me to explicitly solve for $x_B$, $y_B$, $|AB|$, and $|OB|$ in terms of $h$, $\alpha$, and $p$? Let me know! 😊

Relative Questions:

1. How is the focal length $p$ related to the parabola's geometry?
2. What is the effect of varying $\alpha$ on the coordinates of $B$?
3. How does the height $h$ influence the distance $|AB|$?
4. Can $A$ ever coincide with $B$? Under what conditions?
5. What happens to $|OB|$ as $\alpha \to 90^\circ$?

Tip:

When working with parabolas, always start by identifying the vertex, focus, and directrix for setting up the coordinate system.