Step 1: Understanding the Concept:
To ascertain the number of common tangents to a pair of circles, we must determine their geometrical relative position. This is achieved by calculating the distance $d$ between their centers and comparing it against the sum ($r_1 + r_2$) and the absolute difference ($|r_1 - r_2|$) of their radii.
Step 2: Key Formula or Approach:
1. Center and radius of general circle $x^2 + y^2 + 2gx + 2fy + c = 0$ are $(-g, -f)$ and $\sqrt{g^2 + f^2 - c}$.
2. If $d>r_1 + r_2$, circles are strictly outside (disjoint) $\rightarrow 4$ common tangents.
3. If $d = r_1 + r_2$, circles touch externally $\rightarrow 3$ common tangents.
4. If $|r_1 - r_2|<d<r_1 + r_2$, circles intersect $\rightarrow 2$ common tangents.
5. If $d = |r_1 - r_2|$, circles touch internally $\rightarrow 1$ common tangent.
6. If $d<|r_1 - r_2|$, one circle is inside the other $\rightarrow 0$ common tangents.
Step 3: Detailed Explanation:
Let's analyze the first circle, $C_1$:
Equation: $x^2 + y^2 - 6x = 0$
Comparing with standard form, $2g = -6 \implies g = -3$; $2f = 0 \implies f = 0$; $c = 0$.
Center $O_1 = (-g, -f) = (3, 0)$.
Radius $r_1 = \sqrt{(-3)^2 + 0^2 - 0} = \sqrt{9} = 3$.
Let's analyze the second circle, $C_2$:
Equation: $x^2 + y^2 + 6x + 2y + 1 = 0$
Comparing with standard form, $2g = 6 \implies g = 3$; $2f = 2 \implies f = 1$; $c = 1$.
Center $O_2 = (-g, -f) = (-3, -1)$.
Radius $r_2 = \sqrt{3^2 + 1^2 - 1} = \sqrt{9} = 3$.
Now compute the distance $d$ between centers $O_1(3, 0)$ and $O_2(-3, -1)$:
\[ d = \sqrt{(-3 - 3)^2 + (-1 - 0)^2} \]
\[ d = \sqrt{(-6)^2 + (-1)^2} \]
\[ d = \sqrt{36 + 1} = \sqrt{37} \]
Next, calculate the sum of their radii:
\[ r_1 + r_2 = 3 + 3 = 6 \]
Now compare the distance $d$ with the sum of radii $r_1 + r_2$:
We know $6 = \sqrt{36}$.
Since $\sqrt{37}>\sqrt{36}$, it means $d>r_1 + r_2$.
This condition signifies that the two circles are completely disjoint; they lie strictly outside each other without touching or intersecting.
In this configuration, exactly 4 common tangents can be drawn (two direct common tangents and two transverse common tangents).
Step 4: Final Answer:
The number of common tangents is 4.