Conic Sections

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. They are fundamental in describing planetary orbits, parabolic reflectors, and structural arches.

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-

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

Identifying Conics

  • 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

Note

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

Note

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

Standard Forms

Circle

A circle is the set of all points equidistant from a center point (h,k)(h, k).

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

Parabola

A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix).

(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)}

Ellipse

An ellipse is the set of points where the sum of the distances to two focal points is constant.

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

If a>ba \gt b, the major axis is horizontal. If b>ab \gt a, the major axis is vertical.

Hyperbola

A hyperbola is the set of points where the difference of the distances to two focal points is constant.

(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)}

Properties of Conics

Eccentricity (ee)

Eccentricity measures how much a conic section deviates from being circular. It 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 latus rectum is the line segment passing through the focus (or foci) of a conic section, perpendicular to the major axis, with both endpoints on the curve.

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

Note

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)
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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