Important definitions and formulas - Circles - Class 10 PDF Download

⇒ Tangent to a circle:
A line which touches a circle only in one point is called a tangent line and the point at which it touches the circle is called the point of contact.

⇒ Intersection of a line and 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:
(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.

(ii) The line ‘l’ touches lire circle in only one point, Such a line which touches the circle only in one point is called a tangent line and the point at which it touches the circle is called the point of contact.

(iii) 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.

⇒ Important Results and Theorems: 
(i) Number of tangents to a circle from a point: 
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) intersects the circle in two points. Hence, there is no tangent through the point P lies inside the circle.

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

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

(iv) The lengths 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

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

⇒ Theorem-1: 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 tangent to a circle is 90°.
⇒ Theorem-2: The lengths of tangents drawn from an external point to a circle are equal.