Tangent Of A Circle

Before understanding the concept of tanget of a circle, let us understand about the circle

Circle:

A circle is a set of all points in a plane which are equally spaced from a fixed point. The fixed point is called the center of the circle and the distance between any point on the circle and its center is called the radius .

Given, a line to a circle could either be intersecting, non-intersecting or just touching the circle or non-touching.

Consider any line AB and a circle.There are 3 possibilities as shown in the below figures.

Tangent Of A Circle

In Figure-1, line \(AB\) intersects the circle at two points \(P\) and \(Q\). Such a line is called secant of the circle. \(P\) and \(Q\) are the points on the circle; \(PQ\) is a chord of the circle.

In Figure-2, line \(AB\) touches the circle exactly at one point, \(P\). Such a line is called the tangent to the circle.

In Figure-3, line \(AB\) does not touch the circle at any point and is referred to as a non-intersecting line.

Tangent to a circle

A tangent to a circle is a line which intersects the circle at only one point.

The common point between the tangent and the circle is called point of contact .

Imagine a bicycle moving on a road. If we look at its wheel, we observe that it touches the road at just one point. The road can be considered as a tangent to the wheel.

Tangent Of A Circle

It is to be noted that there can one and only one tangent through any given point on the circle.

Any other line through a point on the circle other than the tangent at that point would intersect the circle at two points. This can be easily seen from the following figure.

Tangent Of A Circle

\(\overleftrightarrow{AB}, \overleftrightarrow{CD}, \overleftrightarrow{EF}, \overleftrightarrow{GH}, \overleftrightarrow{IJ}\) are a few lines passing through the point \(P\), where \(P\) is a point on the circle. We observe that all the lines except \(\overleftrightarrow{AB}\) pass through \(P\) and cut the circle at some other point. Hence, only \(\overleftrightarrow{AB}\) is a tangent and \(\overleftrightarrow{CD}, \overleftrightarrow{EF}, \overleftrightarrow{GH}~ and ~\overleftrightarrow{IJ}\) are secants to the circle.

Every secant has a corresponding chord to the circle. Therefore, a tangent can be considered as a special case of secant when the endpoints of its corresponding chord coincide.

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Practise This Question

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is -