Intersection of a line and a circle
Substitute the line into the circle; the discriminant counts the intersection points.
Key formulas
Substitution
From y=mx+c into x²+y²=r² (or the shifted circle), collect a quadratic in x.
Tangent condition
Discriminant zero means exactly one solution — the line is tangent.
Example sketch
x²+y²=25 and y=3: then x²+9=25 → x=±4. Two points (4,3) and (−4,3).
Worked example
Worked example
Problem. Find the intersection points of x² + y² = 25 and the line y = 3.
Solution. x² + 9 = 25 → x² = 16 → x = ±4. Points: (4, 3) and (−4, 3).
Geometry meaning
Distance from centre to line equals radius ⇔ tangent.
Review quiz
x²+y²=16 and y=x. Number of intersection points:
Practice questions
- x²+y²=16 and y=x. Find the points.
- How does discriminant zero appear on the diagram?
Study notes
Must I solve for both x and y?
Yes — each root x gives a matching y from the line.