**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**

**Two intersection points**

**One intersection point**

**One intersection point**

**No intersection points**

**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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