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Intersection points of a circle and a line
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
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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