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

Formula

For a circle x2 + y2 = r2,
if the tangent point on the circle is (x1, y1),

then the equation of the tangent line is
x1x + y1y = r2.

Example

Example

Solution

Draw the conditions.

First draw the circle
x2 + y2 = 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).

x2 + y2 = 10
So r2 = 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.