Question

PQ is tangent to a circle at a point P on the circle. The number of tangents which can be drawn to the circle parallel to PQ, is

• A. 2
• B. 1
• C. many
• D. zero

Hint:

A circle can have a maximum of two parallel tangents at any given time, and these must be at the opposite ends of a diameter.
Since one tangent \(PQ\) is already specified at point \(P\), only 1 more parallel tangent can be drawn at the other end of the diameter.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question explores the properties of tangents to a circle, specifically geometry related to parallel lines and tangents.
We are given a tangent \(PQ\) touching the circle at point \(P\).
We need to find how many other tangents can be drawn to this circle that are parallel to the given tangent \(PQ\).

Step 2: Key Formula or Approach:
Let us consider a circle with centre \(O\). 1. A tangent to a circle is perpendicular to the radius drawn through the point of contact. Therefore, the radius \(OP \perp PQ\).
2. Any line parallel to the tangent \(PQ\) must also be perpendicular to the diameter passing through \(P\).
3. A circle can have at most two parallel tangents at the opposite ends of any given diameter.

Step 3: Detailed Explanation:
1. Let \(P\) be the point of contact of the tangent \(PQ\) on the circle.
2. Draw a diameter through the point \(P\). Let the other endpoint of this diameter be \(R\).
3. Since \(PQ\) is a tangent at \(P\), we know that the diameter \(PR\) is perpendicular to \(PQ\): \[ \angle OPQ = 90^\circ \] 4. For any other tangent to be parallel to \(PQ\), it must also be perpendicular to the diameter \(PR\).
5. The only other point on the circle that lies on this diameter line is the opposite endpoint \(R\).
6. Drawing a tangent at point \(R\) creates a line perpendicular to \(PR\), which makes it parallel to \(PQ\).
7. No other tangent drawn at any other point on the circle can be perpendicular to the diameter \(PR\).
8. Therefore, there is exactly one tangent that can be drawn parallel to the given tangent \(PQ\).

Step 4: Final Answer:
The number of tangents parallel to \(PQ\) is exactly 1.
Hence, the correct option is (B).