Conic Sections: A Complete Tutorial
A comprehensive journey through circles, ellipses, parabolas, and hyperbolas—from foundational concepts to advanced applications.
Formula Quick Reference (click to expand)
Center: \((h,k)\), Radius: \(r\)
\(c^2 = a^2 - b^2\), \(e = c/a \lt 1\)
Focus: \((h, k+p)\), Directrix: \(y = k-p\)
\(c^2 = a^2 + b^2\), \(e = c/a \gt 1\)
\(B^2 - 4AC \lt 0\): Ellipse
\(B^2 - 4AC = 0\): Parabola
\(B^2 - 4AC \gt 0\): Hyperbola
\(e\) = eccentricity, \(d\) = focus-directrix distance
Foundation: Understanding Conic Sections
What Are Conic Sections?
Conic sections are curves obtained by intersecting a plane with a double cone (two identical cones placed vertex-to-vertex). The type of conic section depends on the angle at which the plane intersects the cone relative to the cone's axis.
Historical Context
The Focus-Directrix Definition
A more unified definition uses the concept of eccentricity (\(e\)):
The value of eccentricity determines the type of conic:
| Conic Type | Eccentricity (\(e\)) | Geometric Property |
|---|---|---|
| Circle | \(e = 0\) | All points equidistant from center |
| Ellipse | \(0 \lt e \lt 1\) | Sum of distances to two foci is constant |
| Parabola | \(e = 1\) | Equal distance to focus and directrix |
| Hyperbola | \(e \gt 1\) | Absolute difference of distances to two foci is constant |
Interactive: Eccentricity Morphing
Drag the slider to see how the conic section transforms as eccentricity changes from 0 to 2.
Circle (\(e = 0\))
Definition and Properties
A circle is the set of all points in a plane that are equidistant from a fixed point called the center. This constant distance is called the radius (\(r\)).
Key Features
- Center: \((h, k)\)
- Radius: \(r\)
- Diameter: \(2r\)
- Circumference: \(2\pi r\)
- Area: \(\pi r^2\)
Example: Circle Equations (Both Directions)
These two problems show how the same circle can be expressed in different forms.
Part A - Standard Form from Center and Radius: Find the equation given center \((3, -2)\) and radius \(5\).
Part B - Center and Radius from General Form: Convert \(x^2 + y^2 - 6x + 4y - 12 = 0\) to standard form.
Result: Center \((3, -2)\), Radius \(5\) — the same circle as Part A!
Practice Problems: Circle
- Find the equation of a circle with center \((-4, 5)\) and radius \(7\).
- Convert \(x^2 + y^2 + 8x - 10y + 16 = 0\) to standard form.
- Find the center and radius of \(x^2 + y^2 - 2x + 6y - 15 = 0\).
- Determine if the point \((2, 3)\) lies inside, outside, or on the circle \((x - 1)^2 + (y - 1)^2 = 9\).
Show Solutions
Solution 1: \((x + 4)^2 + (y - 5)^2 = 49\)
Solution 2:
Solution 3:
Center: \((1, -3)\), Radius: \(5\)
Solution 4:
Since \(d \lt r\), the point is inside the circle.
Ellipse (\(0 \lt e \lt 1\))
Definition and Properties
An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points (called foci) is constant.
Standard Forms
Key Features and Formulas
- Center: \((h, k)\)
- Semi-major axis: \(a\) (larger value)
- Semi-minor axis: \(b\) (smaller value)
- Focal distance: \(c\), where \(c^2 = a^2 - b^2\)
- Eccentricity: \(e = \frac{c}{a}\), where \(0 \lt e \lt 1\)
- Vertices: Located \(a\) units from center along major axis
- Co-vertices: Located \(b\) units from center along minor axis
- Foci: Located \(c\) units from center along major axis
Why does \(c^2 = a^2 - b^2\)? (Derivation)
Consider a point \(P\) at the co-vertex \((0, b)\) on the ellipse. By the focal definition, the sum of distances to both foci equals \(2a\).
Interactive: Focus-Directrix Relationship
Drag point P around the ellipse to see that the ratio of (distance to focus) / (distance to directrix) always equals eccentricity e.
Example 3: Finding the Equation of an Ellipse
Problem: Find the equation of an ellipse with center at origin, vertices at \((\pm 5, 0)\), and co-vertices at \((0, \pm 3)\).
Solution:
Example 4: Converting General Form to Standard Form
Problem: Convert \(9x^2 + 4y^2 - 36x + 8y + 4 = 0\) to standard form and identify all key features.
Solution:
Key Features:
- Center: \((2, -1)\)
- Vertical major axis (since \(9 \gt 4\))
- \(a^2 = 9\), so \(a = 3\); \(b^2 = 4\), so \(b = 2\)
- \(c^2 = 9 - 4 = 5\), so \(c = \sqrt{5}\)
- Vertices: \((2, -1 \pm 3) = (2, 2)\) and \((2, -4)\)
- Co-vertices: \((2 \pm 2, -1) = (4, -1)\) and \((0, -1)\)
- Foci: \((2, -1 \pm \sqrt{5})\)
- Eccentricity: \(e = \frac{\sqrt{5}}{3} \approx 0.745\)
Example 5: Using the Focal Definition
Problem: An ellipse has foci at \((0, 3)\) and \((0, -3)\), and the sum of distances from any point on the ellipse to the foci is 10. Find the equation.
Solution:
Practice Problems: Ellipse
- Find the equation of an ellipse with center \((-2, 3)\), horizontal major axis of length 12, and minor axis of length 8.
- Convert \(25x^2 + 9y^2 = 225\) to standard form and find the foci.
- An ellipse has vertices at \((1, 7)\) and \((1, -1)\), and co-vertices at \((-2, 3)\) and \((4, 3)\). Find its equation.
- Find the eccentricity of the ellipse \(\frac{x^2}{64} + \frac{y^2}{100} = 1\).
- Convert \(4x^2 + y^2 - 8x + 4y - 8 = 0\) to standard form.
Show Solutions
Solution 1:
Solution 2:
Solution 3:
Solution 4:
Solution 5:
Historical Context
Parabola (\(e = 1\))
Definition and Properties
A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
Standard Forms
- Vertex at origin
- Opens up if \(p \gt 0\)
- Opens down if \(p \lt 0\)
- Focus: \((0, p)\)
- Directrix: \(y = -p\)
- Vertex at origin
- Opens right if \(p \gt 0\)
- Opens left if \(p \lt 0\)
- Focus: \((p, 0)\)
- Directrix: \(x = -p\)
Key Features
- Vertex: \((h, k)\) - the point closest to the directrix
- Focus: \(p\) units from vertex along axis of symmetry
- Directrix: Line \(p\) units from vertex (opposite side of focus)
- Axis of symmetry: Line through vertex and focus
- Focal parameter \(p\): Distance from vertex to focus (and vertex to directrix)
- Latus rectum: Line segment through focus perpendicular to axis, length = \(|4p|\)
Example 6: Finding Parabola from Focus and Directrix
Problem: Find the equation of a parabola with focus at \((0, 3)\) and directrix \(y = -3\).
Solution:
Example 7: Converting to Standard Form
Problem: Convert \(y^2 - 6y - 8x + 1 = 0\) to standard form and identify all key features.
Solution:
Key Features:
- Vertex: \((-1, 3)\)
- Opens right (horizontal, \(p \gt 0\))
- Focus: \((-1 + 2, 3) = (1, 3)\)
- Directrix: \(x = -1 - 2 = -3\)
- Axis of symmetry: \(y = 3\)
- Latus rectum length: \(|4p| = 8\)
Practice Problems: Parabola
- Find the equation of a parabola with vertex \((2, -1)\) and focus \((2, 3)\).
- Convert \(x^2 + 4x - 12y + 16 = 0\) to standard form and find the focus and directrix.
- A parabola has focus \((-3, 0)\) and directrix \(x = 3\). Find its equation.
- Find the vertex and focus of \(y^2 = -16x\).
Show Solutions
Solution 1:
Solution 2:
Solution 3:
Solution 4:
Historical Context
Hyperbola (\(e \gt 1\))
Definition and Properties
A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from two fixed points (the foci) is constant.
Standard Forms
- Opens left and right
- Vertices: \((\pm a, 0)\)
- Foci: \((\pm c, 0)\)
- Opens up and down
- Vertices: \((0, \pm a)\)
- Foci: \((0, \pm c)\)
Key Features and Formulas
- Center: \((h, k)\)
- Transverse axis: Line segment connecting the vertices, length = \(2a\)
- Conjugate axis: Perpendicular to transverse axis through center, length = \(2b\)
- Vertices: \(a\) units from center along transverse axis
- Foci: \(c\) units from center along transverse axis, where \(c^2 = a^2 + b^2\)
- Eccentricity: \(e = \frac{c}{a}\) (always \(\gt 1\))
-
Asymptotes (center at origin):
- Horizontal: \(y = \pm \frac{b}{a}x\)
- Vertical: \(y = \pm \frac{a}{b}x\)
Example 8: Finding Hyperbola from Vertices and Foci
Problem: Find the equation of a hyperbola with center at origin, vertices at \((0, \pm 4)\), and foci at \((0, \pm 5)\).
Solution:
Example 9: Converting General Form to Standard Form
Problem: Convert \(9x^2 - 16y^2 - 54x - 32y - 79 = 0\) to standard form and identify all key features.
Solution:
Key Features:
- Center: \((3, -1)\)
- Horizontal transverse axis
- \(a^2 = 16\), so \(a = 4\); \(b^2 = 9\), so \(b = 3\)
- \(c^2 = 16 + 9 = 25\), so \(c = 5\)
- Vertices: \((3 \pm 4, -1) = (7, -1)\) and \((-1, -1)\)
- Foci: \((3 \pm 5, -1) = (8, -1)\) and \((-2, -1)\)
- Asymptotes: \(y + 1 = \pm \frac{3}{4}(x - 3)\)
- Eccentricity: \(e = \frac{5}{4} = 1.25\)
Practice Problems: Hyperbola
- Find the equation of a hyperbola with center \((0, 0)\), vertices \((\pm 3, 0)\), and foci \((\pm 5, 0)\).
- Convert \(4y^2 - 9x^2 = 36\) to standard form and find the vertices and foci.
- Find the asymptotes of \(\frac{x^2}{16} - \frac{y^2}{25} = 1\).
- A hyperbola has foci at \((0, -6)\) and \((0, 6)\), and vertices at \((0, -4)\) and \((0, 4)\). Find its equation and eccentricity.
Show Solutions
Solution 1:
Solution 2:
Solution 3:
Solution 4:
Intermediate Techniques
Identifying Conics Using the Discriminant
The general second-degree equation in two variables is:
The type of conic can be determined using the discriminant \(B^2 - 4AC\):
| Discriminant | Conic Type | Additional Conditions |
|---|---|---|
| \(B^2 - 4AC \lt 0\) | Ellipse | If \(A = C\) and \(B = 0\), it's a circle |
| \(B^2 - 4AC = 0\) | Parabola | — |
| \(B^2 - 4AC \gt 0\) | Hyperbola | — |
| Any | Degenerate case | If equation factors into lines or a point |
Example 10: Identifying Conics
Problem: Identify the conic represented by each equation:
(a) \(3x^2 + 2xy + 3y^2 - 4x + 2y - 5 = 0\)
(b) \(x^2 - 4xy + 4y^2 + 2x - y + 1 = 0\)
(c) \(2x^2 - 3xy - 2y^2 + x + 3y - 1 = 0\)
Solution:
(a) \(A = 3\), \(B = 2\), \(C = 3\)
This is an ellipse.
(b) \(A = 1\), \(B = -4\), \(C = 4\)
This is a parabola.
(c) \(A = 2\), \(B = -3\), \(C = -2\)
This is a hyperbola.
Completing the Square
To convert a general second-degree equation (without the \(xy\) term) to standard form, we use the method of completing the square.
- Group \(x\) and \(y\) terms separately
- Factor out coefficients of \(x^2\) and \(y^2\)
- Complete the square for each variable
- Add appropriate values to both sides
- Convert to standard form
Example 11: Complete the Square for an Ellipse
Problem: Convert \(4x^2 + 9y^2 - 16x + 18y - 11 = 0\) to standard form.
Solution:
This is an ellipse with center \((2, -1)\), \(a = 3\) (horizontal), \(b = 2\).
Tangent Lines and Normal Lines
For a conic section, the tangent line at a point touches the curve at exactly one point (locally), while the normal line is perpendicular to the tangent at that point.
- Use implicit differentiation to find \(\frac{dy}{dx}\)
- Evaluate at the point of tangency to get slope
- Use point-slope form: \(y - y_1 = m(x - x_1)\)
For ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the tangent at \((x_0, y_0)\) is:
$$\frac{x \cdot x_0}{a^2} + \frac{y \cdot y_0}{b^2} = 1$$For hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the tangent at \((x_0, y_0)\) is:
$$\frac{x \cdot x_0}{a^2} - \frac{y \cdot y_0}{b^2} = 1$$Example: Tangent Line Using Implicit Differentiation
Problem: Find the equation of the tangent line to the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) at the point \((3\cos\theta, 2\sin\theta)\) where \(\theta = \frac{\pi}{3}\).
Solution:
Verification: Using the shortcut formula \(\frac{x \cdot x_0}{9} + \frac{y \cdot y_0}{4} = 1\): \(\frac{x \cdot \frac{3}{2}}{9} + \frac{y\sqrt{3}}{4} = 1\) gives the same line!
Reflective Properties of Conics
Parabola: Parallel Rays Focus
Parallel rays reflect through the focus.
Ellipse: Focus to Focus
Rays from one focus reflect to the other.
Historical Context
Practice Problems: Intermediate Level
- Use the discriminant to identify: \(x^2 + 4xy + 4y^2 - 3x + 2y = 0\)
- Convert to standard form: \(x^2 - 4y^2 - 2x - 16y - 19 = 0\)
- Convert to standard form: \(9x^2 + 4y^2 + 36x - 8y + 4 = 0\)
- Find the tangent line to \(x^2 + y^2 = 25\) at \((3, 4)\).
- Find the tangent line to \(\frac{x^2}{9} - \frac{y^2}{16} = 1\) at \((5, \frac{16}{3})\).
Advanced Topics
Rotation of Axes
When a conic equation contains an \(xy\) term, the conic is rotated. We can eliminate the \(xy\) term by rotating the coordinate system through an angle \(\theta\).
To eliminate the \(xy\) term in \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), use:
$$\cot(2\theta) = \frac{A - C}{B}$$Example 12: Eliminating the \(xy\) Term
Problem: Eliminate the \(xy\) term from \(xy = 4\) and identify the conic.
Solution:
This is a hyperbola rotated 45° with \(a = b = 2\sqrt{2}\).
Parametric Equations
Parametric equations express \(x\) and \(y\) as functions of a third variable (usually \(t\)).
| Conic | Parametric Equations | Parameter Range |
|---|---|---|
| Circle \(x^2 + y^2 = r^2\) |
\(x = r\cos t\) \(y = r\sin t\) |
\(0 \leq t \lt 2\pi\) |
| Ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) |
\(x = a\cos t\) \(y = b\sin t\) |
\(0 \leq t \lt 2\pi\) |
| Parabola \(y^2 = 4px\) |
\(x = pt^2\) \(y = 2pt\) |
\(-\infty \lt t \lt \infty\) |
| Hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) |
\(x = a\sec t\) \(y = b\tan t\) |
\(0 \leq t \lt 2\pi, t \neq \frac{\pi}{2}, \frac{3\pi}{2}\) |
Example 13: Parametric to Cartesian
Problem: Convert the parametric equations \(x = 3\cos t\), \(y = 5\sin t\) to Cartesian form.
Solution:
This is an ellipse with \(a = 5\) (vertical) and \(b = 3\).
Polar Coordinate Representations
In polar coordinates with focus at the origin, all conics can be represented by a single unified equation.
| Conic Type | Eccentricity | Polar Equation Example |
|---|---|---|
| Circle (special) | \(e = 0\) | \(r = a\) (constant) |
| Ellipse | \(0 \lt e \lt 1\) | \(r = \frac{3}{1 + 0.5\cos\theta}\) |
| Parabola | \(e = 1\) | \(r = \frac{2}{1 + \cos\theta}\) |
| Hyperbola | \(e \gt 1\) | \(r = \frac{6}{1 + 2\cos\theta}\) |
Example 14: Identifying Conic from Polar Equation
Problem: Identify the conic \(r = \frac{12}{3 - 2\cos\theta}\) and find its eccentricity.
Solution:
Since \(0 \lt e \lt 1\), this is an ellipse with \(ed = 4\), so \(d = 6\).
Real-World Applications
- First Law: Planets move in elliptical orbits with the Sun at one focus
- Second Law: A line from the Sun to a planet sweeps out equal areas in equal times
- Third Law: The square of orbital period is proportional to the cube of the semi-major axis
Eccentricity determines orbit shape: Earth (\(e \approx 0.017\)) has nearly circular orbit, while Halley's Comet (\(e \approx 0.967\)) has highly elliptical orbit.
- Satellite dishes: Collect parallel signals and focus them to receiver at focus
- Headlights: Light source at focus produces parallel beam
- Solar cookers: Focus sunlight to achieve high temperatures
- Radio telescopes: Focus radio waves from space
- Elliptical arches: Distribute weight evenly (better than circular arches)
- Whispering galleries: Elliptical domes allow sound to travel from one focus to another
- Cooling towers: Hyperboloid structures provide strength with minimum material
Historical Context
Calculus with Conics
For ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the perimeter is approximately:
$$P \approx \pi(a + b)\left(1 + \frac{3h}{10 + \sqrt{4 - 3h}}\right)$$ where \(h = \frac{(a-b)^2}{(a+b)^2}\)Note: There is no simple closed form for ellipse perimeter; this is Ramanujan's approximation.
- Circle: \(A = \pi r^2\)
- Ellipse: \(A = \pi ab\)
- Parabola (finite region): Use integration
Rotating a parabola \(y = x^2\) from \(x = 0\) to \(x = a\) about the \(y\)-axis creates a paraboloid:
$$S = \frac{\pi}{6}\left[(1 + 4a^2)^{3/2} - 1\right]$$Degenerate Cases
When a plane intersects a double cone in special ways, we get degenerate conics:
| Degenerate Case | Geometric Description | Algebraic Example |
|---|---|---|
| Point | Plane through vertex only | \(x^2 + y^2 = 0\) → point \((0,0)\) |
| Line | Tangent plane to cone | \((x - y)^2 = 0\) → line \(y = x\) |
| Two intersecting lines | Plane through vertex and cone | \(x^2 - y^2 = 0\) → lines \(y = \pm x\) |
| Empty set | No real solutions | \(x^2 + y^2 = -1\) |
Degenerate cases are limiting forms of standard conics. For example, as the eccentricity of a hyperbola approaches infinity, it approaches its asymptotes (two intersecting lines).
Three-Dimensional Extensions: Quadric Surfaces
Conics extend naturally to three dimensions, creating quadric surfaces.
| Surface | Equation | Description |
|---|---|---|
| Sphere | \(x^2 + y^2 + z^2 = r^2\) | 3D extension of circle |
| Ellipsoid | \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\) | 3D extension of ellipse; all cross-sections are ellipses |
| Elliptic Paraboloid | \(z = \frac{x^2}{a^2} + \frac{y^2}{b^2}\) | Bowl-shaped; cross-sections are parabolas and ellipses |
| Hyperbolic Paraboloid | \(z = \frac{x^2}{a^2} - \frac{y^2}{b^2}\) | Saddle-shaped; "Pringles chip" surface |
| Hyperboloid (1 sheet) | \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\) | Cooling tower shape; ruled surface |
| Hyperboloid (2 sheets) | \(\frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\) | Two separate surfaces facing away |
| Elliptic Cone | \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}\) | Double cone; degenerate quadric |
- Hyperboloid cooling towers: Structural strength with minimum material
- Ellipsoid Earth model: More accurate than sphere for GPS
- Paraboloid reflectors: 3D parabolic mirrors and antennas
- Hyperbolic paraboloid roofs: Architecturally striking and structurally efficient
Practice Problems: Advanced Level
- Eliminate the \(xy\) term from \(5x^2 + 6xy + 5y^2 - 8 = 0\) and identify the conic.
- Write parametric equations for the hyperbola \(\frac{x^2}{16} - \frac{y^2}{9} = 1\).
- Identify the conic \(r = \frac{15}{5 - 3\sin\theta}\) and find its eccentricity.
- Find the area enclosed by the ellipse \(\frac{x^2}{25} + \frac{y^2}{16} = 1\).
- Identify the quadric surface: \(x^2 + y^2 - z^2 = 1\).
Comprehensive Comparison of All Conics
| Property | Circle | Ellipse | Parabola | Hyperbola |
|---|---|---|---|---|
| Eccentricity \(e\) | \(e = 0\) | \(0 \lt e \lt 1\) | \(e = 1\) | \(e \gt 1\) |
| Standard Form | \(x^2 + y^2 = r^2\) | \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) | \(x^2 = 4py\) | \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) |
| Number of Foci | 1 (at center) | 2 | 1 | 2 |
| Focal Relationship | — | \(c^2 = a^2 - b^2\) | — | \(c^2 = a^2 + b^2\) |
| Asymptotes | None | None | None | \(y = \pm\frac{b}{a}x\) |
| Directrix | At infinity | 2 lines | 1 line | 2 lines |
| Polar Form | \(r = a\) | \(r = \frac{ed}{1 \pm e\cos\theta}\) | \(r = \frac{d}{1 \pm \cos\theta}\) | \(r = \frac{ed}{1 \pm e\cos\theta}\) |
| Parametric | \(x = r\cos t\) \(y = r\sin t\) |
\(x = a\cos t\) \(y = b\sin t\) |
\(x = pt^2\) \(y = 2pt\) |
\(x = a\sec t\) \(y = b\tan t\) |
| Key Application | Wheels, pipes | Planetary orbits | Projectile motion | Navigation (LORAN) |
Conclusion: The Unity of Conic Sections
Throughout this tutorial, we've explored how conic sections—circles, ellipses, parabolas, and hyperbolas—form a unified family of curves, all arising from the intersection of a plane with a double cone.
From ancient Greek mathematics to modern GPS systems, from Kepler's laws of planetary motion to satellite dish design, conic sections demonstrate the deep connection between pure mathematics and physical reality. The same equations that describe planetary orbits also govern the shape of a thrown ball, the design of telescope mirrors, and the architecture of cooling towers.
As you continue your mathematical journey, remember that these curves represent more than abstract equations—they are fundamental patterns in nature, technology, and art, revealing the mathematical structure underlying our universe.
Definitions: The Coordinate Plane · Conic Sections
Formula quizzes: Coordinate Geometry Formulas · Conic Section Formulas
The synthetic groundwork this page builds on is in the Euclidean Geometry tutorial. Next in the study order is the Number Theory tutorial, which opens Tier 2.