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)
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