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 the circle x2 + y2 = 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: mOP.

mOP = 1/3

Slope of a line

The radius (OP) and the tangent line
are perpendicular.

Tangent to a circle

mOP = 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) = -3x + 9

9 + 1 = 10

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