CIRCLES | SPEED NOTES | MATHEMATICS

Pre-Requisites

Circle:

Circle is a round shaped figure has no corners or edges.

A circle is the locus of all points in a plane which are at constant distance

(called radius) from a fixed point (called centre). A circle with centre O and radius r is denoted by C (O, r).

Radius:

A line segement that joins the centre and circumference or boundary of the circles is called the radius of the circles.

A line segement that divides the circle into two halves is called he diameter of the circle.

Diameter = 2x radius

Radius = Diameter/2

Chord:

A line segment joining any two points on a circle. The largest chord of a circle is a diameter.

Position of A point With Respect To a Circle:

In a plane a point P can lie either inside, or on the circle or outside the given circle.

Position of A Line With Respect to A Circle

If a circle C(O, r) and a straight line ‘l’ are in the same plane, then only three possibilities are there. These are :

Outside The Circle:

(i) The line ‘l’ does not intersect the circle at all. The line ‘l’ is called a non-intersecting line with respect to the circle.

Inside of the Circle – Secant To A Circle:

The line ‘ l ‘ intersects the circle in two distinct points say A and B. The line which intersects the circle in two distinct points is called a secant line.

Touching The Circle – Tangent To Circle:

A tangent to a circle is a special case of the secant when the two end points of the corresponding chord are coincide.

That is the line ‘ l ‘ touches the circle in only one point. Such a line which touches the circle only in one point is called a tangent line.

Tangent To Circle :

Etimology of Tangent:

The word ‘tangent’ comes from the Latin word ‘tangere’, which means to touch and was introduced by the Danish mathematician Thomas Fincke in 1583.

Tangent is a line that intersects the circle in exactly one point.

A tangent to a circle is the limiting position of a secant when its two points of intersection with the circle coincide.

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

In other words the point, at which the tangent touches the circle is called

the point of contact.

Number of Tangents From A Point To Circle:

Number of tangents to a circle from a point (say P) depends upon the position of the point P.

(a)

When point ‘P’ lies outside the circle: There are only two lines, which touch the circle in one point only, all the remaining lines either intersect in two points or do not intersect the, circle. Hence, there are only two tangents from point P to the circle.

(b)

When point ‘ P ‘ lies on the circle : There is only one line which touches the circle in one point, all other lines meet the circle in more than one point. Hence, there is one and only one tangent to the circle through the point P lies on the circle.

(c)

When point ‘ P ‘ lies inside the circle: Every line passing through the point P (lies inside the circle) intersect the circle in two points. Hence, there is no tangent through the point P lies inside the circle

There is only one tangent at a point on the circumference of the circle.

Point of contact is the common point of the tangent and the circle.

The tangent at any point of a circle is perpendicular to the radius through the point of

contact.

Theorems :

(i) Tangent-Radius Theorem

The line perpendicular to the tangent and passing through the point of contact, is known as the normal.

Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

The converse of above theorem is also true.

Theorem :

The tangents at any point of a circle is perpendicular to the radius through the point of contact. Or At the point of contact the angle between radius and tangents to a circle is 90^0 .

Theorem :

The length of tangents drawn from an external point to a circle are equal.

Important Results:

If two circles touch internally or externally, the point of contact lies on the straight line through the two centres.

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

The length of the tangents drawn from an external point to a circle are equal.

Length of the tangent from a point P’ lies outside the circle is given by

PT =PT’ =

The distances between two parallel tangents drawn to a circle is equal to the diameter of the circle.

Facts:

In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.

More Points To Remember !

There is no tangent to a circle passing through a point lying inside the circle.

At any point on the circle there can be one and only one tangent.

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

There are exactly two tangents to a circle through a point outside the circle.

The length of the segment of the tangent from the external point P and the point of contact with the circle is called the length of the tangent.

The lengths of the tangents drawn from an external point to a circle are equal.

The line containing the radius through the point of contact of tangent is called the normal to the circle at the point.

There is no tangent to the circle passing through a point lying inside the cirele.

There are exactly two tangents to a cirele through a point lying outside the circle

The length of the segment of the tangent from the external point and the point of contact

with the circle is called the length of the tangent.

The length of tangents drawn from an external point to a circle are equal.