## Tangents to a circle motivated by chords drawn from points coming closer and closer to the point.

INTRODUCTION
In class IX, we have discussed about the circle and its centre and radius. Recall that a circle is a collection of all the points in a plane which are at a constant distance (called as radius) from a fixed point (called as centre). We have further studied about the chord, the segment, the sector, an arc etc. related to a circle. In this chapter, we shall discuss about the tangent to a circle and its properties. [The word tangent comes from the Latin word “tangere” which means to touch. It was introduced by the Danish Mathematician Thomas Fimeke in 1583.]

circle
A circle is a set of all the points in a plane which are at a constant distance from the fixed point. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle.

A circle with centre at O and radius  = r is generally written as C(O, r).
A line segment formed by joining the two points on the circle and passing through the centre of circle is called the diameter of the circle.

Secant
A line which intersects a circle in two distinct points is called a secant of the circle. PQ is a line which intersects a circle in two distinct points A and B. PQ is a secant.

Tangent To a circle
A tangent to a circle is a line that intersects the circle at only one point. In figure PQ is a tangent to a circle and R is called the point of contact of the tangent. The point of intersection of the circle and a tangent to it is known as point of contact.

Number of Tangents from a point on a circle
Case–I: There is no tangent to a circle passing through a point lying inside the circle.

Case–II: There is one and only one tangent to a circle passing through a point lying on the circle.

Two circles and their common tangents
Case-I: When circles intersect in two points:
In this case, there will be two common tangents PQ and RS to the two circles as shown in figure.

Some properties of Tangent to a Circle

Theorem : 1
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Proof: Let AB be a tangent to the circle with centre O at the point P as shown in the figure. Join OP. We have to prove that OP is perpendicular to AB or AB is perpendicular to OP.
Take a point Q on AB other than the point P. Join OQ. If the point Q lies inside or on the circle, then the line PQ will intersect the circle in two different points and hence a secant.

Which contradicts the tangency of the line.
Therefore, the line Q lies outside the circle.
OQ is greater than the radius OP, i.e. OQ > OP. Since it happens for every point on the line AB except the point P.
Therefore, out of all the line segments joining the centre to any point on the line AB, the line segment OP is the shortest one.
As we know that among all the line segments joining the point O to a point on the line AB, the shortest one is perpendicular to the line AB.

Theorem : 2
A line drawn through the end point of a radius and perpendicular to it is a tangent to the circle.

Proof: Let AB be a line perpendicular to the radius OP of a circle with centre at O as shown in the figure.
Take a point Q other than P on the line AB. Since OP is perpendicular to the line AB.
Out of all the line segments joining O to a point on the line AB, OP is the shortest one.
So,    OP < OQ or OQ > OP
The point Q lies exterior to (or outside) the circle.
Every point other than P on the line AB is an exterior point of the circle.
The line AB meets the circle at only one point P.
Hence, the line AB is a tangent to the circle at the point P.

Length of Tangent
The length of the segment to the tangent between the point and the given points of contact with the circle is called the length of the tangent from the point to the circle. In figure PT and PT’ are the lengths of tangents from point P to the circle.

Normal to a Circle : The line containing the radius through the point of contact is known as the normal to the circle at the point of contact.

IMPORTANT RESULTS FOR CIRCLE AND TANGENTS TO A CIRCLE
1.    One and only one tangent can be drawn at any point on the circle.
2.    If PAB is a secant to a circle intersecting it at A and B and PT is a tangent, then PA × PB = PT2.
3.    The points of intersection of direct common tangents and transverse common tangents to two circles divide the line segment joining the two centres externally and internally respectively in the ratio of their radii.
4.    If two chords AB and CD of a circle intersect each other at P outside the circle, then PA ´ PB = PC ´ PD.