# Equation of a Tangent Line: Circle

How to find the equation of a tangent line to a circle at the given point on the circle: formula, 1 example, and its solution.

## Formula

For a circle x^{2} + y^{2} = r^{2},

if the tangent point on the circle is (x_{1}, y_{1}),

then the equation of the tangent line is

x_{1}x + y_{1}y = r^{2}.

## Example

Draw the conditions.

First draw the circle

x^{2} + y^{2} = 10.

It says

find the line tangent to the circle at (3, 1).

It means

(3, 1) is the tangent point

on the circle.

So draw (3, 1).

And draw the tangent line

that touches the circle at (3, 1).

x^{2} + y^{2} = 10

So r^{2} = 10.

The tangent point is (3, 1).

Then the equation of the tangent line is

3⋅x + 1⋅y = 10.

3⋅x = 3x

+1⋅y = +y

Write the linear equation

in slope-intercept form.

Move 3x to the right side.

Then y = -3x + 10.

So

y = -3x + 10

is the answer.