Conic Sections

Conic Section Overview

Definitions and General Concepts

• Conic Sections: Curves obtained by intersecting a cone with a plane. These include circles, ellipses, parabolas, and hyperbolas.
• Cone: A three-dimensional geometric figure created by rotating a straight line (generator) around another fixed straight line (axis).

Elements of the Cone

• Vertex: The common point where the elements (generating lines) of the cone meet.
• Nappes: The two symmetrical parts of the cone on either side of the vertex.

Types of Conic Sections

• Circle: Formed when the intersecting plane is perpendicular to the cone's axis.
• Ellipse: Occurs when the plane cuts across the cone at an oblique angle, intersecting only one nappe.
• Parabola: Created when the plane intersects the cone parallel to a generating line, cutting only one nappe.
• Hyperbola: Formed when the plane cuts through both nappes parallel to the cone's axis.

Equations of Conic Sections

Circle

• Definition: A set of points equidistant from a fixed point (center).

• Radius: The distance from the center to any point on the circle.

• Standard Equation:
  $(x - h)^2 + (y - k)^2 = r^2$
  where (h, k) is the center, and r is the radius.

• General Equation:
  $x^2 + y^2 + 2gx + 2fy + c = 0$

• Properties:

  1. The coefficients of $x^2$ and $y^2$ are both equal to 1.
  2. The equation has no $xy$ term.

• Center: $(-g, f)$

• Radius:

  1. If c > 0 : $r = \sqrt{g^2 + f^2 - c}$ (Real Circle)
  2. If $c = 0$: Point circle.
  3. If c < 0 : Imaginary Circle.

• Tangent to a Circle: Equation of the tangent at point $(x<em>1, y</em>1)$:
  $x x<em>1 + y y</em>1 = r^2$

• Normal to a Circle: Perpendicular line at the point of tangency.

Properties of the Circle

1. The perpendicular dropped from the center to a chord bisects the chord.
2. Congruent chords are equidistant from the center.
3. The angle in a semi-circle is a right angle.
4. Tangent at any point is perpendicular to the radius at that point.

Parabola

• Definition: A set of points where the distance from each point to a fixed point (focus) is equal to its distance from a fixed straight line (directrix).
• Vertex: Midpoint of the line segment from the focus to the directrix.

Key Features of Parabolas

• Focus: A specific point used in the definition.
• Directrix: A fixed line.
• Latus Rectum: Length of the line segment perpendicular to the axis through the focus.
• Eccentricity: The constant ratio of distances.

Standard Forms

1. Horizontal Parabola:

   • Equation: $y^2 = 4ax$
   • Focus: $(a, 0)$
   • Directrix: $x = -a$

2. Vertical Parabola:

   • Equation: $x^2 = 4ay$
   • Focus: $(0, a)$
   • Directrix: $y = -a$

Theorems on Parabola

1. The point on the parabola closest to the focus is the vertex.
2. The ordinate of any point on the parabola is a mean proportional length of the latus rectum to the abscissa.

Ellipse

• Definition: A set of points where the distance from any point on the ellipse to two foci always equals a constant (length of major axis).
• Standard Forms:

1. Horizontal Ellipse:
   $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
2. Vertical Ellipse:
   $\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1$

Main Characteristics of Ellipses

1. Foci: Points from which distances are measured.
2. Eccentricity: $e = \frac{c}{a}$ (where 0 < e < 1)
3. Lengths:
   • Length of Major Axis: $2a$
   • Length of Minor Axis: $2b$
   • Distance between foci: $2c$

Hyperbola

• Definition: A set of points where the difference of the distances to two fixed points (foci) is constant.
• Standard Forms:

1. Horizontal Hyperbola:
   $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
2. Vertical Hyperbola:
   $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$

Main Characteristics of Hyperbolas

1. Foci: Two fixed points; $c = \sqrt{a^2 + b^2}$
2. Eccentricity: $e = \frac{c}{a}$ (e > 1)
3. Transverse Axis Length: $2a$
4. Conjugate Axis Length: $2b$
5. Asymptotes: Lines approached by the hyperbola but never met.

Theorems on Hyperbola

1. The focal distances satisfy the equation:
   $|PF<em>1| - |PF</em>2| = 2a$
2. The distance between the center and a focus is given by:
   $c = \sqrt{a^2 + b^2}$

Locus

• Definition: A set of points satisfying certain conditions.

Loci in 2-Dimensional Space

1. Circle: Set of all points at a constant distance from a fixed point.
2. Ellipse: Sum of distances from two fixed points is constant.
3. Hyperbola: Difference of distances from two fixed points is constant.
4. Parabola: Points equidistant from a focus and a directrix.
5. Perpendicular bisector: Locus equidistant from two points.

Loci in 3-Dimensional Space

1. Sphere: Points at constant distance from a fixed point in space.
2. Cylinder: Points at constant distance from a fixed line.

Transformations

• Translation of Axis: Change of the coordinate system to simplify equations.
• Rotation of Axis: Removing xy-terms by rotating coordinates.
• Elimination of the xy-term: Achieved through manipulating the angle based on coefficients of the conic's general form.