Conic Sections
Grade 11 · Math · Study Guide
Explore the four conic sections — circle, parabola, ellipse, and hyperbola — formed by slicing a cone with a plane.
Conic sections are curves formed when a plane intersects a double cone. Depending on the angle of the slice, you get a circle, ellipse, parabola, or hyperbola. Each has a standard-form equation that reveals its key features.
A circle with center (h, k) and radius r satisfies (x−h)² + (y−k)² = r². An ellipse has two semi-axes a and b with equation (x−h)²/a² + (y−k)²/b² = 1 (when a > b, the major axis is horizontal).
A parabola with vertex (h, k) opening up/down has equation y = a(x−h)² + k. A hyperbola has two branches: (x−h)²/a² − (y−k)²/b² = 1 (horizontal) or (y−k)²/a² − (x−h)²/b² = 1 (vertical).
| Conic | Standard Equation | Key Feature |
|---|---|---|
| Circle | (x−h)² + (y−k)² = r² | Center (h,k), radius r |
| Parabola (vertical) | y = a(x−h)² + k | Vertex (h,k), opens up if a>0 |
| Ellipse (horiz. major) | (x−h)²/a² + (y−k)²/b² = 1 | a > b; foci on major axis |
| Hyperbola (horiz.) | (x−h)²/a² − (y−k)²/b² = 1 | Branches open left/right |
| Condition | Conic |
|---|---|
| A = C | Circle |
| A and C same sign, A ≠ C | Ellipse |
| A and C opposite signs | Hyperbola |
| A = 0 or C = 0 (not both) | Parabola |
Key terms
- Focus
- A fixed point used to define each conic; parabolas have one, ellipses and hyperbolas have two.
- Vertex
- The turning point of a parabola; also the endpoints of the major axis of an ellipse.
- Asymptote
- Lines a hyperbola approaches but never touches, given by y − k = ±(b/a)(x − h).
- Eccentricity
- A measure of how 'stretched' a conic is; circle: 0, ellipse: 0 < e < 1, parabola: 1, hyperbola: e > 1.
Practice questions
What is the center and radius of the circle (x − 3)² + (y + 2)² = 25?
- Center (3, −2), r = 25
- Center (−3, 2), r = 5
- Center (3, −2), r = 5
- Center (3, 2), r = 5
Show answer
Standard form (x−h)²+(y−k)²=r²: h=3, k=−2, r=√25=5.
The equation y = 2(x − 1)² + 3 is a parabola. What is the vertex?
- (1, 3)
- (−1, 3)
- (1, −3)
- (2, 3)
Show answer
Vertex form y = a(x−h)² + k gives vertex (h, k) = (1, 3).
Identify the conic: x²/16 + y²/9 = 1. What are the lengths of the semi-axes?
Show answer
Both terms positive with sum = 1 → ellipse. a² = 16 → a = 4; b² = 9 → b = 3.
Which equation represents a hyperbola opening left and right?
- x²/9 + y²/4 = 1
- x²/9 − y²/4 = 1
- (x−1)² + (y+2)² = 9
- y = x² − 3
Show answer
The subtracted term is y², meaning branches open horizontally (left/right).
Find the foci of the ellipse x²/25 + y²/9 = 1.
Show answer
a² = 25, b² = 9. c² = a² − b² = 25 − 9 = 16, so c = 4. Foci: (±4, 0).
Fun facts
- Planets orbit the Sun in ellipses, with the Sun at one focus — Kepler's First Law confirmed with conic section math.
- Satellite dishes and car headlights use parabolic shapes because a parabola reflects all signals through its focus to a single point.
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