Precalculus › Study Guide › Conic Sections
Conic Sections Equations
Chapter 11 — Precalculus (Stewart)
Complete reference for all four conic section equations: parabolas with focus and directrix, ellipses with eccentricity and foci, hyperbolas with asymptotes, and circles. Includes completing the square, identifying conics from general form, and real-world applications.
Chapter 11 Practice Problems
30+ questions on all four conic sections
Quick Identification
Given the general form Ax squared plus Bxy plus Cy squared plus Dx plus Ey plus F equals 0 (with B equal to 0), classify by the coefficients of the squared terms:
Parabola
Only x squared OR only y squared (not both)
e.g.: y equals x squared plus 2x minus 3
Circle
Both x squared and y squared, same coefficient, same sign
e.g.: x squared plus y squared equals 25
Ellipse
Both x squared and y squared, same sign, different coefficients
e.g.: 4x squared plus 9y squared equals 36
Hyperbola
Both x squared and y squared with opposite signs
e.g.: x squared minus y squared equals 1
Discriminant Method (works when B is not zero)
For any conic Ax squared plus Bxy plus Cy squared plus Dx plus Ey plus F equals 0, compute the discriminant D equals B squared minus 4AC:
- D is less than 0: ellipse or circle
- D equals 0: parabola
- D is greater than 0: hyperbola
Parabola Equations
A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).
Opens Up or Down
(x minus h) squared equals 4p(y minus k)
- Vertex: (h, k)
- Axis of symmetry: x equals h (vertical)
- Focus: (h, k plus p)
- Directrix: y equals k minus p
- p greater than 0: opens up
- p less than 0: opens down
Opens Left or Right
(y minus k) squared equals 4p(x minus h)
- Vertex: (h, k)
- Axis of symmetry: y equals k (horizontal)
- Focus: (h plus p, k)
- Directrix: x equals h minus p
- p greater than 0: opens right
- p less than 0: opens left
Vertex Form vs. Standard Form
Vertex Form
y equals a(x minus h) squared plus k
Vertex is directly readable as (h, k). The parameter a controls width and direction: a greater than 0 opens up, a less than 0 opens down. To find p, set a equal to 1 divided by 4p, so p equals 1 divided by (4a).
Standard Form (Conics)
(x minus h) squared equals 4p(y minus k)
Used in conic sections context because p appears explicitly. Focus and directrix are found immediately. Convert from vertex form by setting 4p equal to 1 divided by a.
Worked Example: Find focus and directrix
Given: x squared equals 12y. Find the vertex, focus, directrix, and axis of symmetry.
Step 1. Rewrite as (x minus 0) squared equals 4p(y minus 0). Vertex is (0, 0).
Step 2. Match to 4p: 4p equals 12, so p equals 3.
Step 3. Axis of symmetry: x equals 0 (the y-axis, vertical).
Step 4. Focus: (0, 0 plus 3) equals (0, 3).
Step 5. Directrix: y equals 0 minus 3, so y equals negative 3.
Check: p equals 3 is positive, so the parabola opens upward. The focus (0, 3) is above the vertex; the directrix y equals negative 3 is below. Correct.
Worked Example: Write equation from focus and directrix
Focus is at (2, 5). Directrix is x equals negative 4. Write the equation.
Step 1. Directrix x equals negative 4 is a vertical line, so the axis is horizontal and the parabola opens left or right.
Step 2. Vertex is midway between focus and directrix along the axis: h equals (2 plus (negative 4)) divided by 2 equals negative 1. k equals 5 (same as focus). Vertex: (negative 1, 5).
Step 3. p equals distance from vertex to focus: p equals 2 minus (negative 1) equals 3. Positive, so parabola opens right.
Step 4. Equation: (y minus 5) squared equals 4 times 3 times (x minus (negative 1)), which is (y minus 5) squared equals 12(x plus 1).
Ellipse Equations
An ellipse is the set of all points where the sum of distances from two fixed points (foci) is constant. That constant equals 2a (twice the semi-major axis).
Horizontal Major Axis
(x minus h) squared divided by a squared, plus (y minus k) squared divided by b squared, equals 1
where a is greater than b, and a is greater than 0 and b is greater than 0
- Center: (h, k)
- Vertices (ends of major axis): (h plus or minus a, k)
- Co-vertices (ends of minor axis): (h, k plus or minus b)
- Foci: (h plus or minus c, k) where c squared equals a squared minus b squared
Vertical Major Axis
(x minus h) squared divided by b squared, plus (y minus k) squared divided by a squared, equals 1
where a is greater than b (larger denominator under y squared)
- Center: (h, k)
- Vertices (ends of major axis): (h, k plus or minus a)
- Co-vertices (ends of minor axis): (h plus or minus b, k)
- Foci: (h, k plus or minus c) where c squared equals a squared minus b squared
Key Relationships for Ellipses
c squared equals a squared minus b squared
Always subtract — unlike hyperbola
e equals c divided by a
Eccentricity: 0 less than e less than 1
a is greater than b is greater than 0
Major axis always longer than minor
Eccentricity Explained
Eccentricity e equals c divided by a measures how "stretched" the ellipse is:
- e equals 0: perfect circle (foci coincide at center)
- e is close to 0 (say, 0.1): nearly circular orbit
- e equals 0.5: noticeably elongated
- e is close to 1 (say, 0.95): very elongated, comet-like orbit
Earth's orbital eccentricity is approximately 0.017 — nearly circular. Pluto's is about 0.25, making its orbit more noticeably elliptical.
Worked Example: Full ellipse analysis
Given: (x plus 3) squared divided by 25 plus (y minus 1) squared divided by 9 equals 1. Find center, vertices, co-vertices, foci, and eccentricity.
Step 1. Identify: a squared equals 25 (larger), b squared equals 9. So a equals 5, b equals 3. Major axis is horizontal (larger denominator under x).
Step 2. Center: h equals negative 3, k equals 1. Center is (negative 3, 1).
Step 3. Vertices: (negative 3 plus or minus 5, 1) equals (2, 1) and (negative 8, 1).
Step 4. Co-vertices: (negative 3, 1 plus or minus 3) equals (negative 3, 4) and (negative 3, negative 2).
Step 5. c squared equals 25 minus 9 equals 16, so c equals 4. Foci: (negative 3 plus or minus 4, 1) equals (1, 1) and (negative 7, 1).
Step 6. Eccentricity: e equals 4 divided by 5 equals 0.8. Moderately elongated.
Hyperbola Equations
A hyperbola is the set of all points where the absolute value of the difference of distances from two fixed foci is constant. That constant equals 2a. Unlike ellipses, hyperbolas have two separate branches.
Opens Left and Right
(x minus h) squared divided by a squared, minus (y minus k) squared divided by b squared, equals 1
- Center: (h, k)
- Vertices: (h plus or minus a, k)
- Foci: (h plus or minus c, k)
- Transverse axis: horizontal
- Asymptotes: y minus k equals plus or minus (b divided by a)(x minus h)
Opens Up and Down
(y minus k) squared divided by a squared, minus (x minus h) squared divided by b squared, equals 1
- Center: (h, k)
- Vertices: (h, k plus or minus a)
- Foci: (h, k plus or minus c)
- Transverse axis: vertical
- Asymptotes: y minus k equals plus or minus (a divided by b)(x minus h)
Key Relationships for Hyperbolas
c squared equals a squared plus b squared
Always add — unlike ellipse
e equals c divided by a
Eccentricity: e is greater than 1
no restriction on a vs. b
a can equal, exceed, or be less than b
Transverse Axis vs. Conjugate Axis
Transverse Axis
Connects the two vertices. Length equals 2a. Points toward the open direction of the hyperbola. The foci lie on this axis at distance c from center.
Conjugate Axis
Perpendicular to the transverse axis, length equals 2b. Used to draw the asymptote rectangle: a rectangle with half-sides a and b centered at (h, k). The asymptotes are the diagonals of this rectangle.
Memory Trick: Asymptote Slopes
For the horizontal hyperbola, x squared is first, so slope is plus or minus b over a (y over x). For the vertical hyperbola, y squared is first, so slope is plus or minus a over b (flipped). Think of it as: the variable that is NOT subtracted determines which axis the hyperbola opens along, and the slope is the "other letter over a" for horizontal, "a over other letter" for vertical.
Worked Example: Full hyperbola analysis
Given: (y minus 2) squared divided by 16, minus (x plus 1) squared divided by 9, equals 1. Find center, vertices, foci, asymptotes, and eccentricity.
Step 1. y squared term is positive and first, so this hyperbola opens up and down. Center: (negative 1, 2).
Step 2. a squared equals 16, b squared equals 9. So a equals 4, b equals 3.
Step 3. Vertices (along vertical axis): (negative 1, 2 plus or minus 4) equals (negative 1, 6) and (negative 1, negative 2).
Step 4. c squared equals 16 plus 9 equals 25, so c equals 5. Foci: (negative 1, 2 plus or minus 5) equals (negative 1, 7) and (negative 1, negative 3).
Step 5. Asymptotes: y minus 2 equals plus or minus (4 divided by 3)(x plus 1). Note slope is a over b for vertical hyperbola.
Step 6. Eccentricity: e equals 5 divided by 4 equals 1.25. Greater than 1, confirming hyperbola.
Circle Equations
A circle is a special ellipse where a equals b equals r (the radius) and eccentricity equals 0.
Standard Form
(x minus h) squared plus (y minus k) squared equals r squared
- Center: (h, k)
- Radius: r (always positive)
- Diameter: 2r
- r squared is on the right side — do not forget to take the square root for r
General Form
x squared plus y squared plus Dx plus Ey plus F equals 0
Convert to standard form by completing the square on both x and y groups.
Result: r squared equals h squared plus k squared minus F (after completing the square)
Worked Example: Circle from general form
Given: x squared plus y squared minus 6x plus 4y minus 12 equals 0. Find center and radius.
Step 1. Group: (x squared minus 6x) plus (y squared plus 4y) equals 12.
Step 2. Complete the square for x: half of negative 6 is negative 3, and (negative 3) squared equals 9. Add 9 to both sides.
Step 3. Complete the square for y: half of 4 is 2, and 2 squared equals 4. Add 4 to both sides.
Step 4. (x minus 3) squared plus (y plus 2) squared equals 12 plus 9 plus 4 equals 25.
Step 5. Center: (3, negative 2). Radius: r equals 5.
Completing the Square for Conics
Completing the square converts a conic from general form to standard form, revealing the center, vertices, and foci. See the full technique at Completing the Square.
General Procedure
- 1. Move the constant to the right side.
- 2. Group x terms together and y terms together.
- 3. Factor out the coefficient of each squared term (if it is not 1).
- 4. Inside each group, add (half the linear coefficient) squared. Add the same amount (times any factored coefficient) to the right side.
- 5. Factor each group as a perfect square trinomial.
- 6. Divide both sides by the right-side constant (for ellipses and hyperbolas) or isolate the linear variable (for parabolas).
Worked Example: Ellipse from general form
Convert 9x squared plus 4y squared minus 36x plus 8y plus 4 equals 0 to standard form.
Step 1. Move constant: 9x squared plus 4y squared minus 36x plus 8y equals negative 4.
Step 2. Group: (9x squared minus 36x) plus (4y squared plus 8y) equals negative 4.
Step 3. Factor: 9(x squared minus 4x) plus 4(y squared plus 2y) equals negative 4.
Step 4. Complete the square: 9(x squared minus 4x plus 4) plus 4(y squared plus 2y plus 1) equals negative 4 plus 9 times 4 plus 4 times 1, which is negative 4 plus 36 plus 4 equals 36.
Step 5. Factor: 9(x minus 2) squared plus 4(y plus 1) squared equals 36.
Step 6. Divide by 36: (x minus 2) squared divided by 4, plus (y plus 1) squared divided by 9, equals 1.
Result. Ellipse: center (2, negative 1), a squared equals 9 (vertical), b squared equals 4. Vertical major axis with a equals 3, b equals 2.
Worked Example: Hyperbola from general form
Convert 4x squared minus y squared minus 24x minus 4y plus 28 equals 0 to standard form.
Step 1. Move constant: 4x squared minus y squared minus 24x minus 4y equals negative 28.
Step 2. Group: (4x squared minus 24x) plus (negative y squared minus 4y) equals negative 28.
Step 3. Factor: 4(x squared minus 6x) minus (y squared plus 4y) equals negative 28.
Step 4. Complete the square: 4(x squared minus 6x plus 9) minus (y squared plus 4y plus 4) equals negative 28 plus 4 times 9 minus 4, which is negative 28 plus 36 minus 4 equals 4.
Step 5. Factor: 4(x minus 3) squared minus (y plus 2) squared equals 4.
Step 6. Divide by 4: (x minus 3) squared divided by 1, minus (y plus 2) squared divided by 4, equals 1.
Result. Hyperbola opening left and right: center (3, negative 2), a equals 1, b equals 2. Asymptotes: y plus 2 equals plus or minus 2(x minus 3).
Identifying Conics from Ax squared plus Bxy plus Cy squared plus Dx plus Ey plus F equals 0
The full general second-degree equation may include a cross term Bxy, which indicates a rotation. Most precalculus problems have B equal to 0, but here is the complete picture.
| Condition | Conic | Note |
|---|---|---|
| A equals 0 or C equals 0 (not both) | Parabola | Only one squared term present |
| A equals C, B equals 0 | Circle | Equal coefficients, no rotation |
| A and C have the same sign, A not equal to C, B equals 0 | Ellipse | Same sign, unequal coefficients |
| A and C have opposite signs, B equals 0 | Hyperbola | One positive, one negative |
| B squared minus 4AC less than 0 | Ellipse or Circle | Works when B is not 0 |
| B squared minus 4AC equals 0 | Parabola | Discriminant test |
| B squared minus 4AC greater than 0 | Hyperbola | Discriminant test |
Practice: Classify each equation
- 3x squared plus 3y squared minus 6x plus 9 equals 0 → A equals C, B equals 0: Circle
- x squared plus 4y squared minus 2x equals 0 → A not equal to C, same sign, B equals 0: Ellipse
- x squared minus 4y squared minus 2x equals 0 → Opposite signs, B equals 0: Hyperbola
- y squared minus 3x plus 4 equals 0 → Only y squared present: Parabola
Real-World Applications
Conic sections appear throughout physics, engineering, astronomy, and everyday technology.
Parabola Applications
- Satellite dishes and radio telescopes: Incoming parallel signals reflect off the parabolic surface and converge at the focus, where the receiver is placed.
- Car headlights and flashlights: The bulb sits at the focus; parabolic reflector sends light in parallel beams forward.
- Suspension bridges: Under uniform load, the main cable forms a parabola (not a catenary).
- Projectile motion: The path of a thrown object (ignoring air resistance) is a parabola.
Ellipse Applications
- Planetary orbits (Kepler's First Law): Every planet orbits the Sun in an ellipse, with the Sun at one focus.
- Whispering galleries: Sound from one focus reflects off the elliptical ceiling and converges at the other focus. A whisper is heard across the room.
- Lithotripsy: Shock waves generated at one focus of an ellipsoidal reflector converge on a kidney stone at the other focus, shattering it without surgery.
- Optical lenses: Ellipsoidal mirrors focus light from one focal point to the other.
Hyperbola Applications
- LORAN navigation: A ship determines its position from the difference in arrival times of radio signals from two fixed transmitters — a hyperbola.
- Cassegrain telescopes: A secondary hyperbolic mirror reflects light from the primary parabolic mirror through a hole to the eyepiece.
- Cooling towers: The cross-section of a hyperboloid cooling tower is a hyperbola, giving structural strength with minimal material.
- Sonic booms: The intersection of a cone of shock waves with the ground traces a hyperbola.
Circle Applications
- Circular motion: Satellites in circular orbits, rotating wheels, clocks — all governed by the circle equation.
- Cell towers: Coverage area is modeled as a circle with the tower at center and radius equal to signal range.
- Radar: A target at known distance from a single station lies on a circle. Two stations narrow it to two points; three stations determine position exactly.
Complete Reference Table
| Property | Parabola | Circle | Ellipse | Hyperbola |
|---|---|---|---|---|
| Standard Form | (x-h)^2 = 4p(y-k) | (x-h)^2 + (y-k)^2 = r^2 | (x-h)^2/a^2 + (y-k)^2/b^2 = 1 | (x-h)^2/a^2 - (y-k)^2/b^2 = 1 |
| Center | Vertex (h, k) | (h, k) | (h, k) | (h, k) |
| Foci Relationship | Focus at p from vertex | No foci (e = 0) | c^2 = a^2 - b^2 | c^2 = a^2 + b^2 |
| Eccentricity | e = 1 | e = 0 | 0 less than e less than 1 | e greater than 1 |
| Asymptotes | None | None | None | y - k = ±(b/a)(x - h) |
| Branches | 1 | 1 (closed) | 1 (closed) | 2 |
Frequently Asked Questions
What is the difference between the focus and the vertex of a parabola?
The vertex is the turning point of the parabola — the point where it changes direction. The focus is a point inside the parabola, located p units from the vertex along the axis of symmetry. The directrix is p units on the other side of the vertex. Every point on the parabola is equidistant from the focus and from the directrix.
How do I know which variable goes with a squared in an ellipse?
The larger denominator is always a squared. If the larger denominator is under x squared, the major axis is horizontal and the vertices are to the left and right. If the larger denominator is under y squared, the major axis is vertical and the vertices are above and below. When the denominators are equal you have a circle.
Why does c squared equal a squared plus b squared for a hyperbola but a squared minus b squared for an ellipse?
This comes from the geometry. For an ellipse, the foci are inside the ellipse, and c is smaller than a — so you subtract to find c. For a hyperbola, the foci are outside the curve, past the vertices, so c is larger than a — you add to find c. Think of it this way: ellipse foci are squeezed inward (subtract), hyperbola foci are pushed outward (add).
Can I have negative r squared in the circle equation?
No. If after completing the square you get r squared equal to a negative number, the equation has no real solution — there is no circle. If r squared equals 0, the graph is a single point (a degenerate circle). Only when r squared is positive do you have an actual circle.
What is the relationship between conic sections and slicing a cone?
Each conic comes from slicing a double cone with a plane at different angles. A slice parallel to one side of the cone creates a parabola. A slice at a steeper angle (not parallel to any side) creates an ellipse or circle. A slice that cuts both cones creates a hyperbola. This is why they are called conic sections — they are literally sections cut from a cone.