CBSE · Class 6-12

Class 10 · Mathematics (Standard)

Unit 10 · Class 10Mathematics (Standard)

Circles

Tangents to a circle, number of tangents from a point, tangent-radius property.

Lessons

6 concepts

1
What is a Tangent?
A tangent to a circle is a line that touches the circle at exactly one point (the point of contact). A secant intersects the circle at two points. As a secant rotates about a fixed point, in the limiting position it beco…
2
Tangent-Radius Perpendicularity
The tangent at any point of a circle is perpendicular to the radius drawn to the point of contact. Formally: if $OT$ is the radius to the point of contact $T$ and $\ell$ is the tangent at $T$ , then $OT\perp\ell$ .
3
Number of Tangents
From a point inside the circle: no tangent. From a point on the circle: exactly one tangent (the tangent at that point). From a point outside the circle: exactly two tangents, both of equal length.
4
Length of Tangent
If an external point $P$ is at distance $d$ from the centre of a circle of radius $r$ , the length of each tangent from $P$ is $\sqrt{d^2 - r^2}$ (by Pythagoras on the right triangle with the radius perpendicular to the t…
5
Angle Between Tangents
If two tangents from an external point $P$ touch the circle at $A$ and $B$ , then $\angle APB + \angle AOB = 180^\circ$ (where $O$ is the centre). The line $OP$ bisects both angles $\angle APB$ and $\angle AOB$ .
6
Common Tangents to Two Circles
Two circles can have: 0 tangents (one inside the other, non-touching), 1 (internally touching), 2 (intersecting), 3 (externally touching), or 4 (non-intersecting, apart). Direct common tangents don't cross the line joini…

Skills to practise

Skill · medium

Circles — Practice