# Equation of a Tangent Line from a Point on the Circle

How to find the equation of a tangent line from a point on the circle: example and its solution.

## Example

Draw the circle *x*^{2} + *y*^{2} = 10

on the coordinate plane.

Its center is (0, 0).

Equation of a circle

Draw *P*(3, 1) on the circle.

Draw the line tangent to the circle at *P*(3, 1).

Your goal is to find

the linear equation of the tangent line.

To find the slope of the tangent line,

first find the slope of *OP*: *m*_{OP}.*m*_{OP} = 1/3

Slope of a line

The radius (*OP*) and the tangent line

are perpendicular.

Tangent to a circle*m*_{OP} = 1/3

Set the slope of the tangent line as *m*.

Then (1/3)⋅*m* = -1.

Perpendicular lines

Multiply 3 on both sides.

Then *m* = -3.

The slope of the tangent line is -3.

And the tangent line passes through *P*(3, 1).

So the tangent line is*y* = -3(*x* - 3) + 1.

Point-slope form

-3(*x* - 3) = -3*x* + 9

9 + 1 = 10

So the tangent line is*y* = -3*x* + 10.