For two non intersecting and non concentric circles in a plane, what is the maximum possible number of common tangents that can be drawn to both circles?

Introduction / Context:
This conceptual question checks understanding of tangents to circles and how two circles in a plane can share common tangents. It is a standard topic in geometry and appears frequently in aptitude and competitive examinations. The classification of tangents depends on whether the circles intersect, touch each other, or lie completely separate from one another. The case described here involves two circles that do not intersect and are not one inside the other in a concentric way.

Given Data / Assumptions:

• There are two circles in a plane.
• The circles are non intersecting and not concentric.
• We need the maximum possible number of common tangents.
• A common tangent is a line that touches both circles.

Concept / Approach:
The number of common tangents for two circles depends on their relative positions. If the circles intersect at two points, there are two common tangents. If the circles touch each other externally, there are three common tangents. If one circle lies completely inside the other without touching, there are no common tangents. When the two circles are separate and do not intersect and one is not inside the other, the maximum number of common tangents is four, consisting of two direct tangents and two transverse tangents.

Step-by-Step Solution:
Step 1: Visualise two circles that are apart from each other, so that the distance between their centres is greater than the sum of their radii.Step 2: From this configuration, note that there are two lines that touch both circles externally on the same side. These are called direct common tangents.Step 3: There are also two lines that cross between the circles and touch each circle on opposite sides. These are called transverse or indirect common tangents.Step 4: Counting all these, we get a total of four distinct common tangents.Step 5: No further distinct tangent line can touch both circles in this configuration, so four is the maximum possible number.

Verification / Alternative check:
To verify, one can draw several cases: intersecting circles, touching circles, and widely separated circles. In the intersecting case, it is possible to draw only two common tangents. In the externally touching case, one more tangent appears at the point of contact, giving three common tangents. In the case where circles are completely separate, geometric construction clearly shows two direct and two transverse tangents, confirming that four is the maximum number. No additional line can be tangent to both circles at different points without coinciding with one of the four constructed tangents.

Why Other Options Are Wrong:

• 2 corresponds to intersecting circles, not to the non intersecting separate case described here.
• 3 corresponds to circles that touch externally, which is a different special position.
• 6 is impossible because there are not enough geometric positions for six distinct lines to touch each circle exactly once.
• 0 would only occur if one circle was strictly inside another without touching, which contradicts the intended maximum case.

Common Pitfalls:

• Confusing intersecting circles with separated circles and giving 2 as the answer.
• Remembering only the case of touching circles and selecting 3 without considering the fully separated case.
• Thinking that if the circles are far apart, tangents cannot be drawn, which is incorrect.

Final Answer:
The maximum number of common tangents that can be drawn to two non intersecting and non concentric circles is 4.