Q2(iii):

Fill in the blanks : (iii) A circle can have __________ parallel tangents at the most.

Given: A circle in a two-dimensional Euclidean plane.

To Find: The maximum number of parallel tangents that a circle can have.

Visual Representation:

Step 1: Understanding the definition of a tangent
A tangent to a circle is a line that intersects the circle at exactly one point. By the property of circles, a tangent at any point of a circle is perpendicular to the radius through the point of contact.

Step 2: Analyzing parallel lines in relation to a circle
Let $L_1$ be a tangent to the circle at point $P$. Let $O$ be the center of the circle. By the tangent-radius theorem, the radius $OP$ is perpendicular to $L_1$ ($OP \perp L_1$).

Step 3: Determining the existence of a parallel tangent
If we draw a line $L_2$ parallel to $L_1$, for $L_2$ to also be a tangent, it must be perpendicular to the same radius $OP$ at a different point. The only point on the circle that lies on the line passing through $O$ and $P$ other than $P$ is the point $Q$, which is the other end of the diameter passing through $P$.

Step 4: Logical Deduction
Since a diameter is a straight line passing through the center, the tangents drawn at the two endpoints of a diameter are always parallel to each other. If we attempt to draw a third tangent $L_3$ parallel to $L_1$ and $L_2$, it would have to be perpendicular to the diameter $PQ$. However, there are no other points on the circle where a tangent can be drawn that is perpendicular to the diameter $PQ$ other than the points $P$ and $Q$ themselves.

Conclusion:
A circle can have only two tangents that are parallel to each other, specifically those drawn at the opposite ends of any diameter.

Final Answer: A circle can have 2 parallel tangents at the most.