这道题怎么解，并详细告诉我---```plain text
20. 已知双曲线 C: x^2/a^2 - y^2/b^2 = 1 的图像经过点 (2, 3)，点 A、F_2 分别是双曲线 C 的左顶点和右焦点。设过 F_2 的直线 l 交 C 的右支于 P、Q 两点，其中点 P 在第一象限。
(1) 求双曲线 C 的标准方程:
(2) 若直线 AP、AQ 分别交直线 x = 1/2 于 M、N 两点，证明: 向量 MF_2 · 向量 NF_2 为定值;
(3) 是否存在常数 λ, 使得 ∠PF_2A = λ∠PAF_2 恒成立? 若存在, 求出 λ 的值; 若不存在, 说明理由。
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**Diagram Description:**
* **Type:** Coordinate plane with a hyperbola and intersecting lines.
* **Coordinate System:** X-axis and Y-axis intersect at the origin O. Positive directions are indicated by arrows.
* **Hyperbola:** A hyperbola C is shown with its two branches (left and right) centered at the origin. Asymptotes are indicated by dashed lines passing through the origin.
* **Points:**
* O: Origin (0, 0).
* A: Located on the negative x-axis, identified as the left vertex.
* F_2: Located on the positive x-axis, identified as the right focus.
* P: Located on the upper part of the right branch of the hyperbola, in the first quadrant.
* Q: Located on the lower part of the right branch of the hyperbola.
* M: Located on a vertical line, intersection of line AP with this vertical line.
* N: Located on a vertical line, intersection of line AQ with this vertical line.
* **Lines:**
* A line l passes through F_2 and intersects the right branch of the hyperbola at P and Q. This line is not explicitly labeled 'l' in the diagram but is implied by the problem description.
* Line segment AP connecting points A and P.
* Line segment AQ connecting points A and Q.
* A vertical line is shown with points M and N on it. This line corresponds to x = 1/2.
* Line segments connecting P, F_2, and A, forming triangle PF_2A.
* Asymptotes of the hyperbola (dashed lines).
* Axes (solid lines).
* **Annotations:**
* Axes are labeled 'x' and 'y'.
* Points A, O, F_2, P, Q, M, N are labeled.
* The vertical line is implicitly x = 1/2.
* **Relative Position:** A is on the negative x-axis, O is at the origin, F_2 is on the positive x-axis. P is in the first quadrant, Q is below the x-axis. M and N are on a vertical line to the right of the origin.
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