Circle Properties Analysis

We are given two circles:

• Circle 1 ($C_1$): $x^2 + y^2 + 2011x + 2012y + 2013 = 0$
• Circle 2 ($C_2$): $x^2 + y^2 + 2011x + 2012y + 2014 = 0$

The question asks about tangents related to these circles. We first analyze the geometric relationship between $C_1$ and $C_2$.

Identifying Circle Centers

The standard equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$.

For both $C_1$ and $C_2$, we have:

• $2g = 2011 \implies g = \frac{2011}{2}$
• $2f = 2012 \implies f = 1006$

The center for both circles is $O = (-\frac{2011}{2}, -1006)$. This means the circles are concentric.

Comparing Circle Radii

The radius squared ($r^2$) is calculated using $r^2 = g^2 + f^2 - c$.

For $C_1$: $r_1^2 = (\frac{2011}{2})^2 + (1006)^2 - 2013$ For $C_2$: $r_2^2 = (\frac{2011}{2})^2 + (1006)^2 - 2014$

Let $K = (\frac{2011}{2})^2 + (1006)^2$. Then $r_1^2 = K - 2013$ and $r_2^2 = K - 2014$. Since $2013 < 2014$, it follows that $-2013 > -2014$. Therefore, $r_1^2 > r_2^2$, which means $r_1 > r_2$.

Determining Common Tangents

The circles share the same center $O$, and the radius $r_2$ of $C_2$ is smaller than the radius $r_1$ of $C_1$ ($r_2 < r_1$). This indicates that Circle $C_2$ lies entirely inside Circle $C_1$.

When one circle is completely inside another, they do not intersect or touch. Consequently, no common tangents can be drawn between them.

Thus, the correct conclusion is that no common tangent can be drawn.