Math's: Meaning of Tangents and Secants: Tangents as a Limiting Case

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Meaning of Secants and Tangents

The terms secants and tangents define the relationship between a line and a circle.

Tangent- A line which touches a circle at exactly one point is called a tangent. The point where it touches the circle is called the point of contact. In the following figure, is a tangent of the circle at , which is called the point of contact.

Secants and Tangents

Secants and Tangents

Secant- A line which intersects the circle in two distinct points is called a secant. In the below figure, is a secant to the circle and and are called the points of intersection of the line and the circle with centre .

Secants and Tangents

Secants and Tangents

Tangent as a Limiting Case

Limiting Case

Limiting Case

  • A tangent is the limiting position of a secant when the two points of intersection coincide.

  • In the adjacent figure, secant intersects the circle in the points and . If a point A, which lies on the circle, of the secant is fixed and the secant rotates about , intersecting the circle at and ultimately attains the position of the line , then it becomes tangent to the circle at .

Tangent and Radius through the Point of Contact

Point of Contact

Point of Contact

A radius, though the point of contact of tangent to a circle, is perpendicular to the tangent at that point. If is a tangent to the circle, with centre , at the point and and are also points on , then being the shortest distance from to will be perpendicular to . That .

Tangents from a Point Outside the Circle

Point Outside the Circle

Point Outside the Circle

  • From an external point, two tangents can be drawn to a circle. If is a point in the exterior of the circle with center and and are some of the lines drawn from point , then only two of them are tangent to the circle and that are and .

  • The lengths of two tangents from an external point are equal. That is

Point Outside the Circle

Point Outside the Circle

The tangents drawn from an external point to a circle are equally inclined to the line joining the point to the center of the circle. That is and

1. In the adjoining figure, and radius of the circle is . Find the length of the tangent from to the circle, with centre .

Point Outside the Circle

Point Outside the Circle

Solution-

In right , we have

Or

Or

i.e. the length of tangent .

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