To solve this problem, we need to analyze the relationship between the given circles and the chord $PQ$ that intersects the circle $\Omega$.

Step 1: Understanding the Geometry

• $\Omega$ is the large circle with a diameter $AB = 5$.
• $\omega_1$ and $\omega_2$ are smaller circles internally tangent to $\Omega$ at points $A$ and $B$ respectively.
  • $\omega_1$ has a radius $r_1 = 1$.
  • $\omega_2$ has a radius $r_2 = 2$.
• $P$ and $Q$ are the points where the common chord of $\omega_1$ and $\omega_2$ intersects $\Omega$.

Step 2: Using Power of a Point Theorem

The key to solving this problem is applying the power of a point theorem. The power of a point $P$ with respect to a circle is given by $|PA \cdot PB|$ for any point $P$ outside the circle.

Step 3: Coordinate System Setup

Let's set up a coordinate system:

• Place $A$ at $(-\frac{5}{2}, 0)$ and $B$ at $(\frac{5}{2}, 0)$.
• The center $O$ of $\Omega$ is at the origin $(0, 0)$.

For the smaller circles:

• The center of $\omega_1$ is at $A_1$ located at $(-\frac{3}{2}, 0)$ with a radius of 1.
• The center of $\omega_2$ is at $A_2$ located at $(\frac{1}{2}, 0)$ with a radius of 2.

Step 4: Distance Between Centers of $\omega_1$ and $\omega_2$

The distance $d$ between the centers $A_1$ and $A_2$ is: $d = |A_1 A_2| = \left|\frac{1}{2} - \left(-\frac{3}{2}\right)\right| = 2$

Step 5: Length of Common Chord PQ

To find the length of the common chord $PQ$, we use the fact that the length of the common chord between two intersecting circles is given by: $PQ = 2 \sqrt{r_1^2 - \left(\frac{d^2 + r_1^2 - r_2^2}{2d}\right)^2}$ Substituting the values: $PQ = 2 \sqrt{1^2 - \left(\frac{2^2 + 1^2 - 2^2}{2 \cdot 2}\right)^2}$ Simplifying further: $PQ = 2 \sqrt{1 - \left(\frac{4 + 1 - 4}{4}\right)^2} = 2 \sqrt{1 - \left(\frac{1}{4}\right)^2} = 2 \sqrt{1 - \frac{1}{16}} = 2 \sqrt{\frac{15}{16}} = \frac{2 \cdot \sqrt{15}}{4} = \frac{\sqrt{15}}{2}$

Final Answer

The length of $PQ$ is $\sqrt{15}$.

Let me know if you want more details or have any questions. Here are some related questions you might be interested in:

1. How does the power of a point theorem apply to this situation?
2. What are the properties of the common chord of two intersecting circles?
3. How can the equation of a circle be used to find the length of a chord?
4. What happens to the common chord's length if the radii of the circles change?
5. How does the internal tangency of circles affect the geometry of the system?

Tip: When dealing with circle problems, always consider the power of a point theorem as it is a powerful tool for finding distances and lengths related to chords and tangents.