生成这个题目的讲解---**Question 10** **Question Stem:** 椭圆 C: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a>b>0)$ 的左顶点为 A, 点 P,Q 均在 C 上, 且关于 y 轴对称. 若直线 AP, AQ 的斜率之积为 $-\frac{1}{4}$, 则的离心率为 **Translation of Question Stem (for clarity, not part of the required output but useful for understanding):** Ellipse C: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a>b>0)$ has its left vertex at A. Points P and Q are both on C, and are symmetric with respect to the y-axis. If the product of the slopes of lines AP and AQ is $-\frac{1}{4}$, then what is the eccentricity? **Options:** A. $\frac{\sqrt{3}}{2}$ B. $\frac{\sqrt{2}}{2}$ C. $\frac{1}{2}$ D. $\frac{1}{3}$ **Chart/Diagram Description:** * **Type:** Geometric figure, specifically an ellipse plotted on a 2D Cartesian coordinate system. * **Main Elements:** * **Coordinate Axes:** A horizontal x-axis and a vertical y-axis intersect at the origin. * **Origin:** Labeled 'O' at the intersection of the x and y axes. * **Ellipse C:** An oval shape centered at the origin, with its major axis aligned with the x-axis and its minor axis aligned with the y-axis. * **Point A:** The left-most point of the ellipse on the x-axis (left vertex). * **Point P:** A point located on the upper-left part of the ellipse (in the second quadrant). * **Point Q:** A point located on the upper-right part of the ellipse (in the first quadrant). Points P and Q appear to be symmetric with respect to the y-axis. * **Lines:** * A straight line segment connecting point A to point P. * A straight line segment connecting point A to point Q.