Equation of a Tangent Line from a Point on the Circle

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

Write an equation of the line tangent to the circle x^2 + y^2 = 10 at P(3, 1).

Draw a radius that meets the tangent line at P(3, 1):
OP.

Find the slope of OP:
mOP = 1/3.

OP and the tangent line are perpendicular.

Tangent to a circle

So (m, slope of the tangent)⋅(1/3) = -1

Perpendicular lines

So m = -3.
And the tangent line passes through P(3, 1).

Then y - 1 = -3(x - 3).

Point-slope form