Conic Sections

Learning Objectives

  • Identify the four standard conic sections from general and standard equations.
  • Understand the geometric properties of conics including center, focus, and eccentricity.
  • Determine the equation of a conic given its properties.
  • Analyze the latus rectum and eccentricity for different conics.

Conic sections are the curves formed by the intersection of a plane with a double-napped right circular cone. The four standard conics are circles, ellipses, parabolas, and hyperbolas. These geometric shapes are fundamental in civil engineering applications, such as defining the geometry of parabolic bridge arches, analyzing highway horizontal and vertical curves, and designing hyperbolic cooling towers. They are also widely used in describing planetary orbits and parabolic reflectors.

Focus

A fixed point (or points) on the interior of a conic section used to define its shape. The distance from any point on the conic to the focus is a key property of the curve.

Directrix

A fixed line associated with a conic section. The ratio of the distance from any point on the conic to the focus, to its distance to the directrix, is a constant known as the eccentricity.

Major Axis

The longest diameter of an ellipse, passing through the center and both foci, with endpoints at the vertices.

Minor Axis

The shortest diameter of an ellipse, perpendicular to the major axis and passing through the center.

The General Equation

All conic sections can be expressed by the general second-degree equation:

General Conic Form

The general second-degree equation that describes all conic sections.

Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

Variables

SymbolDescriptionUnit
A,CA, CCoefficients of the squared terms (determine if it is an ellipse, hyperbola, or circle)-
BBCoefficient of the xy term (causes rotation of the conic)-
D,ED, ECoefficients of the linear terms (determine translation of the conic)-
FFConstant term-

Identifying Conics

By analyzing the discriminant (B2−4ACB^2 - 4AC), we can determine the type of conic section (assuming it is non-degenerate):

  • Parabola: B2−4AC=0B^2 - 4AC = 0
  • Ellipse: B2−4AC<0B^2 - 4AC \lt 0 (If A=CA = C and B=0B = 0, it is a circle).
  • Hyperbola: B2−4AC>0B^2 - 4AC \gt 0

General Conic Explorer

Use the interactive explorer below to study general conics of the form Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. Adjust the sliders to see how the discriminant B2−4ACB^2 - 4AC changes the conic type, and see the exact rotation angle θ\theta required to eliminate the cross-product xyxy term.

Second-Degree Conic & Axis Rotation Explorer

General Equation
(2.0)x2+(1.0)xy+(2.0)y2=8(2.0)x^2 + (1.0)xy + (2.0)y^2 = 8
Axis Rotation Angle
θ=45.0∘(0.785 rad)\theta = 45.0^\circ \quad (0.785\text{ rad})
Eliminates the xyxy term through rotation to produce standard form.
Conic TypeEllipse
Discriminant (B2−4AC)(B^2 - 4AC)-15.0

Interactive Conic Visualizer

Adjust the parameters of the general equation or standard forms to see how the cone-plane intersection creates different curves.

Circle

A circle is the set of all points equidistant from a center point (h,k)(h, k). The distance from the center to any point on the circle is the radius rr.

Standard Equation of a Circle

Defines a circle given its center and radius.

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

Variables

SymbolDescriptionUnit
x,yx, yCoordinates of any point on the circle-
h,kh, kCoordinates of the center of the circle-
rrRadius of the circle-

Parabola

A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). The vertex is located at (h,k)(h, k).

Standard Equation of a Parabola

Defines a vertical or horizontal parabola given its vertex and focal length.

(x−h)2=4p(y−k)(Vertical)(x - h)^2 = 4p(y - k) \quad \text{(Vertical)}(y−k)2=4p(x−h)(Horizontal)(y - k)^2 = 4p(x - h) \quad \text{(Horizontal)}

Variables

SymbolDescriptionUnit
x,yx, yCoordinates of any point on the parabola-
h,kh, kCoordinates of the vertex-
ppDirected distance from the vertex to the focus (focal length)-

Ellipse

An ellipse is the set of points where the sum of the distances to two focal points is constant. The center is located at (h,k)(h, k). If a>ba \gt b, the major axis is horizontal. If b>ab \gt a, the major axis is vertical.

Standard Equation of an Ellipse

Defines an ellipse given its center and the lengths of its semi-axes.

(x−h)2a2+(y−k)2b2=1\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1

Variables

SymbolDescriptionUnit
x,yx, yCoordinates of any point on the ellipse-
h,kh, kCoordinates of the center-
aaSemi-major axis (or semi-minor if b > a)-
bbSemi-minor axis (or semi-major if b > a)-

Hyperbola

A hyperbola is the set of points where the difference of the distances to two focal points is constant. The center is located at (h,k)(h, k).

Standard Equation of a Hyperbola

Defines a hyperbola given its center, semi-transverse axis, and semi-conjugate axis.

(x−h)2a2−(y−k)2b2=1(Horizontal Transverse)\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \quad \text{(Horizontal Transverse)}(y−k)2a2−(x−h)2b2=1(Vertical Transverse)\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \quad \text{(Vertical Transverse)}

Variables

SymbolDescriptionUnit
x,yx, yCoordinates of any point on the hyperbola-
h,kh, kCoordinates of the center-
aaSemi-transverse axis-
bbSemi-conjugate axis-

Eccentricity (ee)

A non-negative real number that uniquely characterizes the shape of a conic section. It measures how much the conic section deviates from being circular.

Eccentricity (ee)

Eccentricity is calculated as e=c/ae = c/a, where cc is the distance from the center to the focus, and aa is the distance from the center to the vertex.

  • Circle: e=0e = 0
  • Ellipse: 0<e<10 \lt e \lt 1
  • Parabola: e=1e = 1
  • Hyperbola: e>1e \gt 1

Latus Rectum (LR)

The line segment passing through the focus (or foci) of a conic section, perpendicular to the major axis, with both endpoints on the curve.

Latus Rectum (LR)

The latus rectum helps define the "width" of the conic at its focus.

  • Parabola: Length of LR =4p= 4p
  • Ellipse and Hyperbola: Length of LR =2b2a= \frac{2b^2}{a}

Eccentricity Explorer

Explore how the eccentricity ee defines the shape of conic sections. Adjust ee from 00 up to 22 and trace the geometric relationship between the focus, directrix, and locus of points.

Conic Section Focus-Directrix Explorer

Current Conic

Ellipse

d(P,F)d(P,D)=e=0.70\frac{d(P, F)}{d(P, D)} = e = 0.70
Eccentricity (e)0.70
Circle (e=0)Ellipse (0<e<1)Parabola (e=1)Hyperbola (e>1)
Directrix Position (d)x = 3.0
Selected Angle (θ)60°

Focus-Directrix Verification

Focus F: (0.00, 0.00)
Point P: (0.78, 1.35)
Distance PF: 1.556
Distance PD: 2.222
Ratio PF/PD:0.700 (≈ e)
x = dF (Focus)P
PF (Focus Vector)
PD (Directrix vector)

Focal Distance (Ellipse)

Relationship between semi-major axis, semi-minor axis, and focal distance for an ellipse.

c2=a2−b2c^2 = a^2 - b^2

Variables

SymbolDescriptionUnit
ccDistance from center to focus-
aaSemi-major axis-
bbSemi-minor axis-

Focal Distance (Hyperbola)

Relationship between semi-transverse axis, semi-conjugate axis, and focal distance for a hyperbola.

c2=a2+b2c^2 = a^2 + b^2

Variables

SymbolDescriptionUnit
ccDistance from center to focus-
aaSemi-transverse axis-
bbSemi-conjugate axis-
Key Takeaways

  • The Center (h,k)(h,k): All standard equations use (h,k)(h,k) as the center or vertex, which is determined by (x−h)(x-h) and (y−k)(y-k).

  • Completing the Square: This is the primary algebraic tool used to convert general conic equations into standard form.

  • Ellipse vs Hyperbola: In an ellipse equation, the x2x^2 and y2y^2 terms are added. In a hyperbola, they are subtracted.

  • Focal Distance: For an ellipse, c2=a2−b2c^2 = a^2 - b^2. For a hyperbola, c2=a2+b2c^2 = a^2 + b^2.

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