CON016 · Conic sections · Classes 11–12
Conic intersections
To find where a line meets a conic, substitute the line into the conic's equation and solve the quadratic. The discriminant tells the number of meeting points: 2, 1 (tangent) or 0.
∩Conic intersections
📐 Formula
x² + (mx + c)² = r² ⇒ (1 + m²)x² + 2mcx + c² − r² = 0
✏️ Worked example
Circle x² + y² = 9 and y = x + 1: discriminant 68 > 0, so 2 points.
🌍 Where it is used
Collision detection in games, ray tracing.
❓ Quick check
A line touching a circle at exactly one point is a…
🏋️ Practice
EasyDoes y = 5 meet x² + y² = 9?
Answer: No
MediumWhere does y = 0 meet x² + y² = 4?
Answer: (±2, 0)
ChallengeFor which c is y = c tangent to x² + y² = 9?
Answer: c = ±3