Two circles with radii 5 cm and 8 cm touch each other externally at a point A. If a straight line through the point A cuts the circles at points P and Q respectively, then AP : AQ is

Options

A. `8 : 5`
B. `5 : 8`
C. `3 : 4`
D. `4 : 5`

Answer

To solve the problem, we need to find the ratio of the segments AP and AQ, where P and Q are points on two circles that touch each other externally at point A. The radii of the circles are given as 5 cm and 8 cm.

### Step-by-Step Solution:

1. **Identify the Circles and Their Centers:**
- Let the first circle (smaller) have a radius \( R_1 = 5 \) cm and center \( O \).
- Let the second circle (larger) have a radius \( R_2 = 8 \) cm and center \( O' \).
- The circles touch each other externally at point \( A \).

2. **Determine the Distance Between the Centers:**
- The distance between the centers \( O \) and \( O' \) is the sum of the radii of the two circles:
\[ OO' = R_1 + R_2 = 5 \, \text{cm} + 8 \, \text{cm} = 13 \, \text{cm} \]

3. **Draw the Line Through Point A:**
- A straight line through point \( A \) intersects the first circle at point \( P \) and the second circle at point \( Q \).

4. **Determine the Lengths AP and AQ:**
- The length \( AP \) is the distance from point \( A \) to point \( P \) on the first circle. Since \( A \) is on the circumference, \( AP \) is equal to the radius of the first circle:
\[ AP = R_1 = 5 \, \text{cm} \]
- The length \( AQ \) is the distance from point \( A \) to point \( Q \) on the second circle. Similarly, \( AQ \) is equal to the radius of the second circle:
\[ AQ = R_2 = 8 \, \text{cm} \]

5. **Calculate the Ratio AP : AQ:**
- Now, we can find the ratio of \( AP \) to \( AQ \):
\[ AP : AQ = 5 : 8 \]

### Final Answer:
The ratio \( AP : AQ \) is \( 5 : 8 \).