Intersection and Tangency Conditions
The graph and the algebra describe the same event. Each real solution of the substituted quadratic is a point where the line and circle meet.
The discriminant counts intersections
For with , the is . Its sign gives the number of real roots, so it also gives the number of intersections.
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- : two distinct real roots, so the line cuts the circle twice.
- : one repeated root, so the line is tangent to the circle.
- : no real roots, so the line misses the circle.
Turn a graph condition into an inequality
Worked example
Classify against the circle
- Substitute: , so .
- .
- For two intersections, gives . Thus lies between the two boundary values: .
- For tangency, , so . The two signs are the two parallel tangents on opposite sides of the centre.
Two intersections:
Tangent:
No intersection: or
Show that a line is tangent
A “show that” answer must show why there is exactly one intersection. Substitute the line and demonstrate either a repeated factor or a zero discriminant. Do not begin by assuming the line is tangent.
Worked example
Show that is tangent to
- Substitute into the circle.
- Simplify: , hence .
- has one repeated root, so the line and circle meet once.
Select the correct inequality
Find the values of for which cuts, touches or misses the circle .
Show worked answer
Substitution gives , with .
Two intersections: .
Tangent: .
No intersection: or .