# Tangent to a Circle

A line that touches the curve is generally known as a tangent. In this article, let us discuss in detail about the tangent to a circle.

## Tangent to a Circle Definition

Tangent to a circle is the line that touches the circle at only one point. There can be only one tangent at a point to circle. Point of tangency is the point at which tangent meets the circle. Now, let’s prove tangent and radius of the circle are perpendicular to each other at the point of contact.

Consider a circle in the above figure whose centre is O. AB is the tangent to a circle through point C.
Take a point D on tangent AB other than at C and join OD. Point D should lie outside the circle because; if point D lies inside, then AB will be a secant to the circle and it will not be a tangent.

Therefore, OD will be greater than the radius of circle OC. This happens for every point on AB except the point of contact C.
It can be concluded that OC is the shortest distance between the centre of circle O and tangent AB.
Since, the shortest distance between a point and a line is the perpendicular distance between them,
OC is perpendicular to AB.

From the above discussion, it can be concluded that

• Only one tangent can be drawn through any point of the circle.
• We can call the line containing the radius through the point of contact as ‘normal’ to the circle at the point

## Tangent to a circle Equation

Here, the list of the tangent to the circle equation is given below:

• The tangent to a circle equation x2+ y2=aat (x1, y1) is xx1+yy1= a2
• The tangent to a circle equation x2+ y2+2gx+2fy+c =0 at (x1, y1) is xx1+yy1+g(x+x1)+f(y +y1)+c =0
• The tangent to a circle equation x2+ y2=aat (a cos θ, a sin θ ) is x cos θ+y sin θ= a
• The tangent to a circle equation x2+ y2=a for a line y = mx +c is y = mx ± a √[1+ m2]

## When the Point Lies Inside the Circle

Consider the point P inside the circle in the above figure; all the lines through P is intersecting the circle at two points.
It can be concluded that no tangent can be drawn to a circle which passes through a point lying inside the circle.

## When the Point Lies on the Circle

From the figure; it can be concluded that there is only one tangent to a circle through a point which lies on the circle.

## When the Point Lies Outside the Circle

From the above figure, we can say that
There are exactly two tangents to circle from a point which lies outside the circle.

### Tangent to a Circle Example

Example: AB is a tangent to a circle with centre O at point A  of radius 6 cm. It meets the line OB such that OB = 10 cm. What is the length of AB?

We know that AB is tangent to the circle at A.
Since tangent AB is perpendicular to the radius OA,
ΔOAB is a right-angled triangle and OB is the hypotenuse of ΔOAB.
By using Pythagoras theorem,

$OB^2$ = $OA^2~+~AB^2$
$AB^2$ = $OB^2~-~OA^2$
$AB$ = $\sqrt{OB^2~-~OA^2 }$
= $\sqrt{10^2~-~6^2}$ = $\sqrt{64}$ = 8 cm