Conic Sections
Conic Sections
Conic Sections
FEMA: LESSON 6
Conic Sections
• The conic sections are the nondegenerate curves generated by the intersections of a plane with
one or two nappes of a cone.
• For a plane that is not perpendicular to the axis and that intersects only a single nappe, the
curve produced is either an ellipse or a parabola.
Focus
The focus is a fixed point used to define the
parabola. This point is not located on the
parabola, but inside. The focus is denoted by F.
Directrix
The directrix is a straight line in front of the
parabola. We use d to represent the directrix. The
distance between the directrix and the vertex is
the same as the distance between the focus and
the vertex.
Vertex
The vertex of the parabola is its extreme point.
If the parabola opens upwards, the vertex
represents the lowest point in the parabola. If
the parabola opens downwards, the vertex
represents the highest point. In either case, the
vertex is a point that changes the direction of
the parabola. Frequently, the vertex is
represented with the letter V.
Focal length
2p 2p The focal length is the length between the
vertex and the focus.
p
Latus rectum
The latus rectum is a line perpendicular to the
line joining the vertex and the focus and is four
times the length of the focal length.
Axis
The axis of the parabola is a line perpendicular
to the directrix. The axis represents the line of
symmetry of the parabola.
Standard form of Parabola with vertex (h, k)
TYPES OF VERTICAL PARABOLA HORIZONTAL PARABOLA
PARABOLA
Equation (𝑥 − ℎ)2 = 4𝑝 (𝑦 − 𝑘) (𝑦 − 𝑘)2 = 4𝑝(𝑥 − ℎ)
Vertex (h, k) (h, k)
Opening If p>0, opens upward If p>0, opens to the right
If p<0, opens downward If p<0, opens to the left
Directrix Horizontal Vertical
• The fixed points are known as the foci, which are surrounded by the curve. The fixed line is
directrix and the constant ratio is eccentricity of ellipse. Eccentricity is a factor of the
ellipse, which demonstrates the elongation of it and is denoted by ‘e’.
Parts of an Ellipse
• Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the
shorter axis is called the minor axis.
• Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each
endpoint of the minor axis is a co-vertex of the ellipse.
• The center of an ellipse is the midpoint of both the major and minor axes. The axes are
perpendicular at the center.
𝑐𝑒𝑛𝑡𝑒𝑟: (0,0)
𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠: ±8, 0 ; 𝑎 = 8
𝑓𝑜𝑐𝑖: ±5, 0 ; 𝑐 = 5
𝑐 = 𝑎 −𝑏
(−8, 0) (−5, 0) (5, 0) (8, 0)
𝑏 = 𝑎 −𝑐
𝑏 = 8 −5
𝑏 = 64 − 25
𝑏 = 39
Example #2: (−2, 8)
( ) ( )
Graph the ellipse by the equation, + =1
( ) ( ) (−2, 5 + 5)
( ) ( ( ))
(−4, 5) (−2, 5) (0, 5)
𝑐𝑒𝑛𝑡𝑒𝑟: (−2, 5)
𝑎 = 9; 𝑎 = 3
𝑏 = 4; 𝑏 = 2 (−2, 5 − 5)
𝑐 = 𝑎 − 𝑏 ;𝑐 = ± 5
𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠: ℎ, 𝑘 ± 𝑎 ; (−2, 5 ± 3)
𝑐𝑜 − 𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠: ℎ ± 𝑏, 𝑘 ; (−2 ± 2, 5) (−2, 2)
𝑓𝑜𝑐𝑖: ℎ, 𝑘 ± 𝑐 ; (−2, 5 ± 5)
Circle
• A circle is formed when a plane cuts the cone at right angles to its axis.
• The definition of a circle is the set of all points in a plane such that each point in
the set is equidistant from a fixed point called the center.
• The distance from the center is called the radius.
• The distance around the circle is called the circumference.
Equation of a circle
• A circle is an ellipse in which both the
foci coincide with its center. As the foci
are at the same point, for a circle, the
distance from the center to a focus is
zero. This eccentricity gives the circle its
round shape. Thus the eccentricity of any
circle is 0.
(𝑥 − ℎ) +(𝑦 − 𝑘) = 𝑟
(𝑥 − 0) +(𝑦 − 0) = 4
𝑥 + 𝑦 = 16 (0, 0)
Example #2:
Graph the circle with equation (𝑥 + 2) +(𝑦 − 3) = 25.
(𝑥 + 2) +(𝑦 − 3) = 25
(𝑥 − (−2)) +(𝑦 − (3)) = 5
𝑐𝑒𝑛𝑡𝑒𝑟: −2, 3
𝑟𝑎𝑑𝑖𝑢𝑠: 5 (−2, 3)