帮我解题---**Problem 1:**
**Question Stem:**
已知F₁, F₂是椭圆和双曲线的公共焦点, P是它们的一个公共点, 且∠F₁PF₂=π/3, 则椭圆和双曲线的离心率之积的最小值为 ( )
**Options:**
A. √3
B. √3/3
C. √3/2
D. 1
**Problem 2 (真题7):**
**Question Stem:**
如图, F₁, F₂分别是双曲线C: x²/a² - y²/b² = 1 (a, b>0) 的左、右焦点, B是虚轴的端点, 直线F₁B与C的两条渐近线分别交于P, Q两点, 线段PQ的垂直平分线与x轴交与点M, 若|MF₂|=|F₁F₂|, 则C的离心率是 .
**Options:**
A. 2√3/3
B. √6/2
C. √2
D. √3
**Chart/Diagram Description (associated with Problem 2):**
* **Type:** Coordinate plane diagram showing a hyperbola, its foci, asymptotes, and related points and lines.
* **Coordinate Axes:** X-axis and Y-axis intersecting at the origin O. Arrows indicate the positive direction for both axes. The X-axis is labeled 'x' and the Y-axis is labeled 'y'.
* **Hyperbola:** A hyperbola C opening left and right symmetrically about the Y-axis, with vertices on the X-axis.
* **Foci:** Points F₁ and F₂ are on the X-axis. F₁ is to the left of the origin, labeled. F₂ is to the right of the origin, labeled. These are the foci of the hyperbola.
* **Points:**
* O: Origin (0,0).
* B: A point on the positive Y-axis, representing an endpoint of the imaginary axis.
* P: A point in the second quadrant, intersection of line F₁B and one asymptote of the hyperbola.
* Q: A point in the first quadrant, intersection of line F₁B and the other asymptote of the hyperbola.
* M: A point on the positive X-axis, intersection of the perpendicular bisector of PQ and the X-axis.
* **Lines:**
* Asymptotes: Two lines passing through the origin, forming an 'X' shape, representing the asymptotes of the hyperbola.
* Line F₁B: A straight line segment connecting F₁ and B, extending to intersect the asymptotes at P and Q.
* Line segment PQ: Connects points P and Q.
* Perpendicular bisector of PQ: A dashed line that intersects the X-axis at M. The dashed line appears to be perpendicular to the segment PQ.
* **Labels:** Points F₁, F₂, O, B, P, Q, M, x, y are labeled.
* **Relative Position:** F₁ is on the negative x-axis, F₂ and M are on the positive x-axis. B is on the positive y-axis. P is in the second quadrant, Q is in the first quadrant. O is at the origin.