# Intersection points of a circle

### Intersection points of a circle and a line

A line and a circle can have two, one or no intersection points.

Two intersection points. The line goes through the circle.

One intersection point. The line is a tangent of the circle.

No intersection points. The line and circle do not meet.

Example 1

Find the points of intersection of the line and circle below.

We set the line in slope-intercept form

We substitute this into the equation of the circle

Simplify

And we get a quadratic equation

Where the solutions are

The y-coordinates of the intersection points are obtained by placing the x-coordinates in the equation of the line. When x = -4, y = -4 + 4 = 0. When x = 0, y = 0 + 4 = 4. The points of intersection of a line and a circle are then (-4,0) and (0,4).

## No intersection points

Example 2

Find the points of intersection of the circles below

Form a pair of equations and subtract the equations from each other.

We get a line that passes through the intersection of the circles. Now we solve x from the equation.

We substitute x to first circle

And we get a quadratic equation

The solutions to the equation are y = 0 or y = 1

The x-coordinates of the intersection point are obtained by substitute y in the equation of the line.

When y = 0, x = 3 ∙ 0 + 1 = 1

When y = 1, x = 3 ∙ 1 + 1 = 4

The intersection points of the circles are (1,0) and (4,1)

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