Question 4: Identify the number of common tangents that exist between each pair of circles.

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1. Consider two circles with radii $$r_1$$ and $$r_2$$ and with the distance between their centers equal to $$d$$. The number of common tangents depends on the relationship between $$d$$, $$r_1$$, and $$r_2$$. 2. We list the cases below: - **Case 1. Completely separate circles:** If $$ d > r_1 + r_2, $$ then the circles do not intersect and are far apart. In this case there are $$ 4 \text{ common tangents.} $$ - **Case 2. Externally tangent circles:** If $$ d = r_1 + r_2, $$ then the circles touch each other externally. In this configuration there are $$ 3 \text{ common tangents.} $$ - **Case 3. Intersecting circles:** If $$ |r_1 - r_2| < d < r_1 + r_2, $$ then the circles intersect at two points. They have $$ 2 \text{ common tangents.} $$ - **Case 4. Internally tangent circles:** If $$ d = |r_1 - r_2|, $$ then one circle touches the other internally. In this case there is $$ 1 \text{ common tangent.} $$ - **Case 5. One circle lies completely inside the other without touching:** If $$ d < |r_1 - r_2|, $$ then one circle is contained entirely inside the other and there are $$ 0 \text{ common tangents.} $$ 3. In summary, the number of common tangents for each configuration is as follows: - $$d > r_1 + r_2 \rightarrow 4$$ tangents. - $$d = r_1 + r_2 \rightarrow 3$$ tangents. - $$|r_1 - r_2| < d < r_1 + r_2 \rightarrow 2$$ tangents. - $$d = |r_1 - r_2| \rightarrow 1$$ tangent. - $$d < |r_1 - r_2| \rightarrow 0$$ tangents. This analysis provides the answer for the number of common tangents between two circles based on their positions relative to each other. There are 4 common tangents when the circles are completely separate, 3 when they are externally tangent, 2 when they intersect, 1 when one is internally tangent to the other, and 0 when one circle is entirely inside the other without touching.

Question 4: Identify the number of common tangents that exist between each pair of circles.

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