Conic Sections

Learning Objectives

• Understand the geometric derivation of conic sections by intersecting a plane with a double-napped cone, including the conditions for obtaining circles, ellipses, parabolas, and hyperbolas.
• Calculate the equation of a circle given its center $(h, k)$ and radius $r$ using the distance formula.
• Derive the standard equation of a parabola with a given focus and directrix, and identify its axis of symmetry, vertex, and latus rectum.
• Derive the standard equation of an ellipse, understanding its definition as a set of points with a constant sum of distances from two foci, and calculate properties such as major/minor axes, eccentricity, and latus rectum.
• Derive the standard equation of a hyperbola, understanding its definition as a set of points with a constant difference of distances from two foci, and calculate properties such as transverse/conjugate axes, eccentricity, and latus rectum.
• Compare the eccentricity of different conic sections: circle $(e = 0)$, parabola $(e = 1)$, ellipse $(0 < e < 1)$, and hyperbola $(e > 1)$, using the formula $e = \frac{c}{a}$ for ellipses and hyperbolas.
• Calculate the length of the latus rectum for parabolas, ellipses, and hyperbolas, and use it to identify conic parameters.
• Identify the type, axis direction, opening direction, center/vertex position, and standard equation of conics from their equation forms.
• Solve problems involving conic equations or parameters from given focus, directrix, vertex, axis, foci, vertices, and lengths of axes.
• Recognize degenerate conic sections such as points, straight lines, and pairs of intersecting lines when the cutting plane passes through the cone vertex.

Chapter Notes

Conic Sections

• Definition: Curves obtained by intersecting a plane with a double-napped cone.
• Types: Circle, ellipse, parabola, and hyperbola.
• Conditions:
  • Circle: Plane angle $\beta = 90^\circ$.
  • Ellipse: $\alpha < \beta < 90^\circ$.
  • Parabola: $\beta = \alpha$.
  • Hyperbola: $0 \leq \beta < \alpha$.
• Degenerate Cases:
  • Point, straight line, or intersecting lines when the plane passes through the cone vertex.

Circle Equation

• Definition: Set of all points equidistant from a fixed point (center).
• Equation: For center $(h, k)$ and radius $r$, $(x-h)^2 + (y-k)^2 = r^2$.

Parabola and its Properties

• Definition: Set of all points equidistant from a fixed line (directrix) and a fixed point (focus).
• Standard Equation: $y^2 = 4ax$ for vertex at origin and focus at $(a, 0)$.
• Properties:
  • Axis of symmetry along the x-axis.
  • Latus rectum length is $4a$.

Ellipse and its Standard Equation

• Definition: Set of all points where the sum of distances from two foci is constant.
• Standard Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ for foci on x-axis.
• Properties:
  • Major axis length $2a$.
  • Minor axis length $2b$.
  • Eccentricity $e = \frac{c}{a}$ where $c^2 = a^2 - b^2$.
  • Latus rectum length $\frac{2b^2}{a}$.

Hyperbola and its Standard Equation

• Definition: Set of all points where the difference of distances from two foci is constant.
• Standard Equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ for foci on x-axis.
• Properties:
  • Transverse axis length $2a$.
  • Conjugate axis length $2b$.
  • Eccentricity $e = \frac{c}{a}$ where $c^2 = a^2 + b^2$.
  • Latus rectum length $\frac{2b^2}{a}$.

Eccentricity of Conic Sections

• Circle: $e = 0$.
• Parabola: $e = 1$.
• Ellipse: $0 < e < 1$.
• Hyperbola: $e > 1$.

Latus Rectum of Conic Sections

• Parabola: Length $4a$.
• Ellipse: Length $\frac{2b^2}{a}$.
• Hyperbola: Length $\frac{2b^2}{a}$.

Standard Forms and Orientation of Conics

• Circle: $(x-h)^2 + (y-k)^2 = r^2$.
• Parabola: $y^2 = 4ax$ or $x^2 = 4ay$.
• Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
• Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.

Focus, Directrix, Axis and Vertex Problems

• Finding Parameters: Use given focus, directrix, and vertex to derive conic equations.

Degenerate Conic Sections

• Point: When plane cuts at the vertex.
• Line: When plane contains a generator of the cone.
• Intersecting Lines: When plane cuts through both nappes at the vertex.