Give step by step answer of the math problem in the image.---**Extraction Content:**
**Question Stem:**
已知: 如图, 二次函数 $y = -\frac{4}{9}x^2 + 4$ 的图象与 $y$ 轴交于点 $A$, 与 $x$ 轴正半轴交于点 $B$, 点 $P$ 在以 $A$ 点为圆心, 2 个单位长度为半径的圆上, $Q$ 点是 $BP$ 的中点, 连接 $OQ$, 则 $OQ$ 的最小值为.
**Mathematical Formulas:**
$y = -\frac{4}{9}x^2 + 4$
**Other Relevant Text:**
Given: As shown in the figure, the graph of the quadratic function $y = -\frac{4}{9}x^2 + 4$ intersects the $y$-axis at point $A$, intersects the positive $x$-axis at point $B$. Point $P$ is on the circle with center $A$ and radius 2 units. Point $Q$ is the midpoint of $BP$. Connect $OQ$. Find the minimum value of $OQ$.
**Chart Description:**
* **Type:** Coordinate plane showing a parabola and a circle.
* **Main Elements:**
* Coordinate Axes: Horizontal X-axis and vertical Y-axis intersecting at the origin O. Positive directions are indicated by arrows.
* Parabola: A downward-opening curve, symmetric about the Y-axis. It passes through point A on the Y-axis and point B on the positive X-axis.
* Circle: Centered at point A. Point P is shown on the circumference of the circle.
* Points: O (Origin), A (on the Y-axis, vertex of the parabola, center of the circle), B (on the positive X-axis, intersection of the parabola and X-axis), P (on the circle), Q (on the segment BP).
* Segments: Segment BP is shown. Segment OQ is shown. Segment PQ is labeled.
* Labels: O, x, y, A, B, P, Q are labeled on the figure.
* **Relative Position and Direction:** Point A is above the origin on the Y-axis. Point B is to the right of the origin on the X-axis. Point P is shown in the first quadrant, on the circle centered at A. Point Q is shown on the segment BP.
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Let's solve this step by step. First, we need to find the coordinates of points A and B. Point A is where the parabola intersects the y-axis. Setting x equals zero, we get y equals 4, so A is at coordinates zero comma four. Point B is where the parabola intersects the positive x-axis. Setting y equals zero and solving, we get x equals 3, so B is at coordinates three comma zero.
Now we introduce point P, which lies on a circle centered at point A with radius 2. The equation of this circle is x squared plus y minus 4 squared equals 4. Point P can move anywhere along this circle. Let's visualize this by showing P moving around the circle.
Since Q is the midpoint of segment BP, as P moves around the circle centered at A, point Q traces out another circle. This new circle has its center at M, which is the midpoint of segment AB at coordinates 1.5 comma 2. The radius of Q's circle is half the radius of P's circle, so it equals 1. Let's see how Q moves as P travels around its circle.
Now we need to calculate the distance from the origin O to the center M of the circle on which Q moves. Using the distance formula, OM equals the square root of 1.5 squared plus 2 squared. This gives us the square root of 2.25 plus 4, which equals the square root of 6.25, or 2.5. Since this distance is greater than the radius of circle Q, which is 1, the origin lies outside the circle.
To find the minimum value of OQ, we use the fact that the shortest distance from a point to a circle is the distance to the center minus the radius. Since OM equals 2.5 and the radius of circle Q is 1, the minimum value of OQ is 2.5 minus 1, which equals 1.5 or three halves. This occurs when Q is at the point on the circle closest to the origin.