Traps in Circles

When converting from general form to standard form, students skip or incorrectly complete the square, leading to wrong center and radius.

Why: Students rush through the algebra and forget to add the same constant to both sides, or mix up signs when factoring.

Students write $S_1 + S_2 = 0$ instead of $S_1 - S_2 = 0$ for the radical axis, or forget to equate the coefficients of $x^2$ and $y^2$ to 1 first.

Why: Confusion between the radical axis (difference) and the family of circles (sum with parameter). Also, the general equations must have coefficient 1 for x^2 and y^2 before subtracting.

Students use the wrong sign when applying the tangency condition $c^2 = a^2(1 + m^2)$, losing one of the two tangent solutions.

Why: The condition comes from setting the discriminant to zero, and students forget that c = +/- a*sqrt(1+m^2) gives two tangent lines.

Students mix up the conditions for internal and external common tangents, especially when counting the number of tangents.

Why: The conditions depend on the relationship between the distance between centers (d) and the sum/difference of radii (r1+r2 and |r1-r2|), which is easy to confuse.

Students confuse signs when extracting center and radius from the general equation $x^2 + y^2 + 2gx + 2fy + c = 0$.

Why: The center is $(-g, -f)$, not $(g, f)$. The sign flip is a consistent source of errors, especially when g and f are already negative.

From an external point, there are exactly two tangent lines. Students often find one and stop.

Why: The quadratic in m (slope) gives two solutions, but students either miss the second root or forget that one tangent may be vertical (infinite slope).