Geometry: Circles
Intersections of a line and a circle
A line and a circle can have two, one, or no points of intersection. In the case that they have one the line is called a tangent line to the circle.
In the figure we see a line and a circle. Move the circle by dragging the center and change the radius by adjusting the slider. Move the line by dragging the two points. See what happens to the number of intersections.
The block describes how to find the intersection of a line and a circle.
Intersections line and circle
Step-by-step |
Example |
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We determine the intersections of a line and a circle . |
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Step 1 |
Write the equation of line in the form |
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Step 2 |
Substitute the equation of line , as found in step 1, in the equation of circle . |
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Step 3 |
expand the brackets of quadratic equation in from step 2 and determine the discriminant
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We find two points of intersection. |
step 4 |
Determine the coordinates of the intersection points by solving the quadratic equation in step 2 using the quadratic formula. |
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step 5 |
Substitute the coordinates found in step 4 in the equation of the line from step 1 to determine the corresponding coordinates. |
If , If , |
Step 1 | The line is already in the form . |
Step 2 | We substitute the equation of line in the equation of the circle. That gives: This can be simplified to: |
Step 3 | We reduce the equation from step 2 to and expand the brackets. This goes as follows: We now read , and for the quadratic formula. That gives: , and . We can now calculate the discriminant. Since the discriminant is equal to , there are solutions. |
Step 4 | We continue to solve the equation using the quadratic formula. |
Step 5 | Now we determine the appropriate -values found by substituting the -values in the equation of line . For , it holds that . For , it holds that . The points of intersection are therefore: |
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