# 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). ### Intersection points of two circles ## Two intersection points ## One intersection point ## 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) Turn on the subtitles if needed