How many parallel tangents may a circle have?

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Question from Manish, a parent:

how many parallel tangents may a circle have? the text book shows two.but a circle can have infinite tangents.then why not parallel tangents coz
theoretically each tangent have a parallel tangents then no. of parallel tangent a circle may have is equals to half of the infinity i.e. infinity..

Manish,

Your are right - there are an infinite number of tangents to a circle, and each of these has a second, parallel tangent (on the 'opposite side' of the circle).

The book seems to be answering a different question:
Given one tangent, how many other tangents are their parallel to this initial tangent?

In math (and some other areas like computer science) a source of confusion can be: which terms are 'Universal Quantifiers' (For All ... ) and which are 'Existential Quantifiers' (There Exists .... )

Here the question seems to unpack as:
Given an initial tangent to a circle, how many other tangents are parallel?

You are answering the question:
How many pairs of parallel tangents are there, for a given circle?

Both questions have answers (and different answers!) so it is a matter of figuring out which question to answer!

Walter Whiteley

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.