Conic Sections

Conics

Conic sections are plane curves that can be formed by cutting a double right circular cone with a plane at various angles.

DEFINITION

DOUBLE RIGHT CIRCULAR CONE

A circle is formed when the plane intersects one cone and is perpendicular to the axis

AXIS

An ellipse is formed when the plane intersects one cone and is NOT perpendicular to the axis.

A parabola is formed when the plane intersects one cone and is parallel to the edge of the cone.

A hyperbola is formed when the plane intersects both cones.

DEGENERATE CONIC

In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2.

It can be defined as the locus of points whose distances are in a fixed ratio to some point, called a focus, and some line, called a directrix.

GENERAL EQUATION OF CONICS

𝑨𝒙𝟐+𝑩𝒙𝒚+𝑪𝒚𝟐+𝑫𝒙+𝑬𝒚+𝑭=𝟎

𝑩𝟐−𝟒 𝑨𝑪<𝟎𝑩𝟐−𝟒 𝑨𝑪=𝟎𝑩𝟐−𝟒 𝑨𝑪>𝟎

Ellipse

Parabola

Hyperbol

a

DISCRIMINANT

Parabola: A = 0 or C = 0 Circle: A = C Ellipse: A = B, but both have the

same sign Hyperbola: A and C have Different

signs

The parabola is a set of points which are equidistant from a fixed point (the focus) and the fixed line (the directrix).

The Parabola

PROPERTIES

The line through the focus perpendicular to the directrix is called the axis of symmetry or simply the axis of the curve.

The point where the axis intersects the curve is the vertex of the parabola. The vertex (denoted by V) is a point midway between the focus and directrix.

The undirected distance from V to F is a positive number denoted by |a|.

The line through F perpendicular to the axis is called the latus rectum whose length is |4a|. The endpoints are and. This determines how the wide the parabola opens.

The line parallel to the latus rectum is called the directrix.

𝑷 (𝒙 , 𝒚 )

Axis of Symmetry

DirectrixLatus Rectum \4a\

VertexFocus

𝑳𝟏

𝑳𝟐

|a|

TYPES OF PARABOLA

𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 :𝒚 𝟐=𝟒𝒂𝒙𝒐𝒑𝒆𝒏𝒊𝒏𝒈 :𝒕𝒐𝒕𝒉𝒆 𝒓𝒊𝒈𝒉𝒕𝒂𝒙𝒊𝒔 :𝒙 𝑽 (𝟎 ,𝟎)

𝑭 (𝒂 ,𝟎)𝑳𝟐(𝒂 ,−𝟐𝒂)𝑳𝟏(𝒂 ,𝟐𝒂)

𝑫 : 𝒙=−𝒂

TYPE 1

𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 :𝒚 𝟐=−𝟒𝒂𝒙𝒐𝒑𝒆𝒏𝒊𝒏𝒈 :𝒕𝒐𝒕𝒉𝒆 𝒍𝒆𝒇𝒕𝒂𝒙𝒊𝒔 :𝒙 𝑽 (𝟎 ,𝟎)

𝑭 (−𝒂 ,𝟎)𝑳𝟐(−𝒂 ,−𝟐𝒂)𝑳𝟏(−𝒂 ,𝟐𝒂)

𝑫 : 𝒙=𝒂

TYPE 2

2L

)2,( aa

𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 :𝒙𝟐=𝟒𝒂𝒚𝒐𝒑𝒆𝒏𝒊𝒏𝒈 :𝒖𝒑𝒘𝒂𝒓𝒅𝒂𝒙𝒊𝒔 :𝒚 𝑽 (𝟎 ,𝟎)

𝑭 (𝟎 ,𝒂) 𝑳𝟐(−𝟐𝒂 ,𝒂)𝑳𝟏(𝟐𝒂 ,𝒂)

𝑫 : 𝒚=−𝒂

TYPE 3

𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 :𝒙𝟐=−𝟒𝒂𝒚𝒐𝒑𝒆𝒏𝒊𝒏𝒈 :𝒅𝒐𝒘𝒏𝒘𝒂𝒓𝒅𝒂𝒙𝒊𝒔 :𝒚 𝑽 (𝟎 ,𝟎)

𝑭 (𝟎 ,−𝒂 )𝑳𝟐(𝟐𝒂 ,−𝒂)𝑳𝟏(−𝟐𝒂 ,−𝒂)

𝑫 : 𝒚=𝒂

TYPE 4

Sample Problem

1.Locate the coordinates of the vertex (V), focus (F), endpoints of the latus rectum (), the equation of the directrix, and sketch the graph of .

solution1. takes the form 2. the parabola opens downward3. Compute the value of 4. so, , or 5. the required coordinates are𝑽 (𝟎 ,𝟎)𝑭 (𝟎 ,−𝒂)=𝑭 (𝟎 ,−𝟐)

𝑳𝟐 (𝟐𝒂 ,−𝒂 )=𝑳𝟐(𝟒 ,−𝟐)𝑳𝟏 (−𝟐𝒂 ,−𝒂 )=𝑳𝟏(−𝟒 ,−𝟐) 𝑫 : 𝒚=𝟐

𝑫 : 𝒚=𝒂

𝑳𝟏(−𝟒 ,−𝟐)

𝒙

𝑳𝟐(𝟒 ,−𝟐)

𝒚=𝟐

𝑭 (𝟎 ,−𝟐)

𝑽 (𝟎 ,𝟎)

𝒚

| | | | | 1 2 3 4 5

| | | | | -5 -4 -3 -2 -1

|

|

|

1

2

3

|

|

|

-3

-

2

-1

Sketch the graphs and determine the coordinates of V, F, ends of LR, and equation of the directrix.

1. 2. 3.