MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.

1. CONIC
SECTIONSPREPARED BY: ROQUI MABUGAY
GONZAGA

2. When a straight line intersect a vertical line at the fixed point and rotate about the
fixed point. The surface obtained is called a double right circular cone.

3. A double right circular cone consist of two cones joined at the fixed point called
the vertex. A line that rotates about the vertex is called the generator. The line that
remain fixed is called the axis. The right circular cone has a circular base and its
axis is always perpendicular to its base.
Vertex
Generator
Lower nappe
Directrix
Upper nappe
Axis
The Perimeter of its base is what we
called the directrix. And the Lateral
surface of the cone is called a nappe.
A double right circular cone has two
nappe, the cone above the vertex is the
upper nappe and below the vertex is the
lower nappe.

4. What is Conics?
What is Conic
Section?

5. What is Conic Section?
 two-dimensional figure created by the intersection of a plane and
a right circular cone.
 If a plane intersects a right circular cone, we get two dimensional
curves of different types. This curves is called the conic sections.

6. What is Conics?
 two-dimensional figure created by the intersection of a
plane and a right circular cone.
There are many types of curve produced when a plane slices through a
cone.
 Parabola
 Circle
 Ellipse
 Hyperbola

7. Types of Conics Section
1. Circle
 Is made from a plane intersecting a cone parallel to its base.
A circle is the locus of points that are
equidistant from a fixed point (the
center).

8. Types of Conics Section
1. Circle
 Let C be a given point. The set of all points P having the same
distance from C is called a circle. The point C is called the center
of the circle and the common distance is its radius.
Q
The circle has a center, C(0,
0) and radius r>0. A point
P(x, y) is on the circle if and
only if 𝑃𝐶 = 𝑟
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛
𝑥2
+ 𝑦2
= 𝑟2
, r > 0
c
P(x,y)

9. Types of Conics Section
1. Circle
 Let C be a given point. The set of all points P having the same
distance from C is called a circle. The point C is called the center
of the circle and the common distance is its radius.
The circle has a center, C(h,
k) and radius r>0. A point
P(x, y) is on the circle if and
only if 𝑃𝐶 = 𝑟
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛
(𝑥 − ℎ)2
+(𝑦 − 𝑘)2
= 𝑟2
, r > 0

10. Types of Conics Section
Example 1:Write the equation of a circle which point (-6, 4) lies
on the circle and center at (-5, 0).
1st Step: Find the radius (𝑥 − ℎ)2
+(𝑦 − 𝑘)2
= 𝑟2
center: (-5, 0), radius: (-6, 0)
---Apply Direct Substitution---
(𝑥 − ℎ)2
+(𝑦 − 𝑘)2
= 𝑟2
(−6 − (−5))2
+(4 − 0)2
= 𝑟2
(−6 + 5)2
+(4)2
= 𝑟2
(−1)2
+16 = 𝑟2
1+16 = 𝑟2
17= 𝑟2
2nd Step: Write the
Standard equation
(𝑥 − ℎ)2
+(𝑦 − 𝑘)2
= 𝑟2
(𝑥 + 5)2
+(𝑦 − 0)2
= 17
Or
(𝑥 + 5)2
+𝑦2
= 17

11. Types of Conics Section
2. Parabola
 Is made from a plane intersecting a cone at an angle parallel to
the slant edge.

12. Types of Conics Section
2. Parabola
 Let F be a given point, and ℓ a given line not containing F. The
set of all points P such that its distances from F and from ℓ are
the same, is called a parabola.
Pℓ
Directrix
P(x, y)
ℓ
F

13. Types of Conics Section
2. Parabola
 A set of points on the coordinate plane that are of equal
distance from a fixed point and line. The fixed point is called
focus and the fixed line is called the directrix.
Pℓ
Directrix
P(x, y)
ℓ
F

14. Types of Conics Section
2. Parabola
The line connecting two points on the parabola and passing through
the focus is called the latus rectum. The Axis of symmetry is the line
which divides the parabola into two equal parts and passes through the
vertex and the focus.
Pℓ
Directrix
P(x, y)
ℓ
F

15. Types of Conics Section
2. Parabola
Vertices Foci Directrices Equation Description
1. (ℎ, 𝑘) (ℎ ± 𝑝, 𝑘) 𝑥 = ℎ − 𝑝 (𝑦 − 𝑘)2= 4𝑝(𝑥 − ℎ) The axis of symmetry is 𝑦 = 𝑘
Open to the right if p>0
Open to the left if p<0
2. (ℎ, 𝑘) (ℎ, 𝑘 ± 𝑝) 𝑦 = 𝑘 − 𝑝 (𝑥 − ℎ)2= 4𝑝(𝑦 − 𝑘) The axis of symmetry is 𝑥 = ℎ
Opens upward if p>0
Opens downward if p<0
Standard Equations of Parabola with vertex at (h, k) and axis of symmetry
Parallel to a coordinate axis
The general Equation of a parabola is;
𝐴𝑥2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0, 𝐸 ≠ 0 𝑖𝑓 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎 𝑜𝑝𝑒𝑛𝑠 𝑢𝑝𝑤𝑎𝑟𝑑/𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑
𝐴𝑥2
+ 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0, 𝐷 ≠ 0 𝑖𝑓 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎 𝑜𝑝𝑒𝑛𝑠 𝑠𝑖𝑑𝑒𝑤𝑎𝑦𝑠

16. Types of Conics Section
Example 1: Identify the coordinates of the vertex, focus, and the equations
of the axis of symmetry and directrix. Then Sketch the graph.
1. (𝑥 − 2)2
= 4 𝑦 − 1
opening of parabola: Upward p>0
4𝑝 = 4
4 = 4
P= 1
Vertex: (2, 1)
Focus: (h, k+p)—(2, 1+1)—(2, 2)
Axis of Symmetry: x=h—(x=2)
Directrix: 𝑦 = 𝑘 − 𝑝 =1-1
y=0
h,
k
x=2
F (2, 2)
V (2, 1)
y = 0

17. Types of Conics Section
3. Ellipse
 Is made from a plane intersecting a cone at an angle parallel
to the slant edge.

18. Types of Conics Section
3. Ellipse
 Let 𝐹1 and 𝐹2 be two distinct points. The set of all points P,
Whose distances from 𝐹1 and from 𝐹2 add up to a certain
constant, is called an ellipse. The points 𝐹1 and 𝐹2 are called
the foci of the ellipse.
𝐹1 𝐹2
𝑃1
𝑃2
𝑷 𝟏 𝑭 𝟏 + 𝑭 𝟏 𝑷 𝟐 = 𝑷 𝟐 𝑭 𝟐 + 𝑭 𝟐 𝑷 𝟏

19. Types of Conics Section
3. Ellipse
 Let 𝐹1 and 𝐹2 be two distinct points. The set of all points P,
Whose distances from 𝐹1 and from 𝐹2 add up to a certain
constant, is called an ellipse. The points 𝐹1 and 𝐹2 are called
the foci of the ellipse.
Vertices Foci Endpoint
of Minor
Axis
Equation Descriptio
n
Directrices Axis of
symmetry
(ℎ ± 𝑎, 𝑘) (ℎ ± 𝑐, 𝑘) (ℎ ± 𝑏, 𝑘) (𝑥 − ℎ)2
𝑎2
+
(𝑦 − 𝑘)2
𝑏2
a>b
Major Axis
is
Horizontal
𝑥 = ℎ ±
𝑎
𝑒
Both Axis
(ℎ, 𝑘 ± 𝑎) (ℎ, 𝑘 ± 𝑐) (ℎ, 𝑘 ± 𝑏) (𝑥 − ℎ)2
𝑏2 +
(𝑦 − 𝑘)2
𝑎2
Major Axis
is Vertical
𝑦 = 𝑘 ±
𝑎
𝑒
Both Axis
Standard Equation of Ellipse with Center (h, k)

20. Types of Conics Section
3. Ellipse
Eccentricity: 𝑒 =
𝑐
𝑎
e(a constant)=
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑓𝑜𝑐𝑢𝑠
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑑𝑖𝑟𝑒𝑐𝑡𝑟𝑖𝑥
e=0 for Circle
0<e<1 for Ellipse
e=1 for Parabola
e>1 for hyperbola
The general form of the equation of an ellipse is
𝐴𝑥2
+ 𝐶𝑦2
+ 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
with AC>0 and a≠ 0.
Properties:
• a, b, c
• Center
• Vertex
• Covertex( endpoints of
Minor axis)
• Foci
• Directrix
• Axis of Symmetry
• Major Axis
• Length of Major Axis
• Minor Axis
• Length of Minor Axis

21. Types of Conics Section
Example 1: Identify the properties of the equations and sketch the graph.
1.
(𝑥−2)2
16
+
(𝑦−1)2
4
= 1 a=4, b=2, c= 12
Center(h, k): 2, 1)
Vertex (h±𝑎, 𝑘): (2 ±4, 1)
𝑉1: 6, 1 & 𝑉2: −2, 1
Foci (h ±c, k): (2± 12, 1)
𝐹1: 2 + 12, 1 or (5.46, 1)
& 𝐹2: 2 − 12, 1 or −1.46, 1
Covertex (h, k±𝑏): (2, 1±2)
𝐵1: 2, 3 & 𝐵2: 2, −1
Directrix 𝑥 = ℎ ±
𝑎
𝑒
: 𝑋1: 2 +
4
12
4
; 2+4.61=6.61
𝑋2: 2 −
4
12
4
; 2-4.61=-2.61
Axis of Symmetry: x=2; y=1
Major Axis: Horizontal
Minor Axis: Vertical
𝐹1 𝐹2
c
𝐵1
𝐵2
𝑉1 𝑉2

22. Types of Conics Section
4. Hyperbola
 Is made from a plane intersecting both halves of a double cone,
but not passing through the apex.

23. Types of Conics Section
4. Hyperbola
Let 𝐹1 and 𝐹2 be two distinct points. The set of all points P,
Whose distances from 𝐹1 and 𝐹2 differ by a certain constant, is
called a hyperbola. The points 𝐹1 and 𝐹2 are called the foci of the
hyperbola.
𝐹1 𝐹2
𝑃1
𝑃2
𝑭 𝟏 𝑷 𝟏 − 𝑭 𝟐 𝑷 𝟐 = 𝑭 𝟏 𝑷 𝟐 − 𝑭 𝟐 𝑷 𝟐

24. Types of Conics Section
4. Hyperbola
A hyperbola is a set of all points in the plane such that the
absolute value of the difference of the distances from two fixed
points are called the foci of the hyperbola.
𝐹1 𝐹2
𝑃1
𝑃2
𝑭 𝟏 𝑷 𝟏 − 𝑭 𝟐 𝑷 𝟐 = 𝑭 𝟏 𝑷 𝟐 − 𝑭 𝟐 𝑷 𝟐

25. Types of Conics Section
4. Hyperbola
Transverse Axis of the parabola is the line that connects the vertices
and has the length of 2a.
Conjugate Axis is the line that connects the co-vertices and has a
length of 2b
𝐹1 𝐹2
𝑃1
𝑃2
𝑭 𝟏 𝑷 𝟏 − 𝑭 𝟐 𝑷 𝟐 = 𝑭 𝟏 𝑷 𝟐 − 𝑭 𝟐 𝑷 𝟐

26. Types of Conics Section
4. Hyperbola
Equation Vertices Foci Endpoints
of
conjugate
axis
Asymptotes Directrices
(𝑥 − ℎ)2
𝑎2
−
𝑦 − 𝑘 2
𝑏2
= 1
𝑏2 = 𝑐2 − 𝑎2
𝑐2
= 𝑎2
+ 𝑏2
(ℎ ± 𝑎, 𝑘) (ℎ ± 𝑐, 𝑘) (ℎ, 𝑘 ± 𝑏)
𝑦 − 𝑘 = ±
𝑏
𝑎
(𝑥 − ℎ) 𝑥 = ℎ ±
𝑎
𝑒
(𝑦 − 𝑘)2
𝑎2
−
𝑥 − ℎ 2
𝑏2
= 1
𝑏2
= 𝑐2
− 𝑎2
(ℎ, 𝑘 ± 𝑎) (ℎ, 𝑘 ± 𝑐) (ℎ ± 𝑏, 𝑘)
𝑦 − 𝑘 = ±
𝑎
𝑏
(𝑥 − ℎ) 𝑦 = 𝑘 ±
𝑎
𝑒
Standard Equation of Ellipse with Center (h, k)

27. Types of Conics Section
4. Hyperbola
Steps in graphing the Hyperbola:
1. Find the vertices of the hyperbola
2. Draw the fundamental triangle
3. Sketch the asymptote as diagonals
4. Sketch the graph. Each graph goes through the
vertex and approaches each asymptotes.

28. Types of Conics Section
Example 1: Sketch the graph:
𝑥2
52 −
𝑦2
42=1
Center: (0, 0)
Vertex (h±𝑎, 𝑘)= (0±5, 0)
𝑉1: 5, 0 & 𝑉2: −5, 0
F(h±𝑐, 𝑘): F(0± 41, 0)
𝐹1: 41 ,0 & 𝐹2: − 41, 0
B(h, k±𝑏): (0, 0±4)
𝐵1: 0, 4 & 𝐵2: 0, −4
Assymptotes: 𝑦 − 𝑘 = ±
𝑏
𝑎
𝑥 − ℎ
𝑦 − 0 = ±
4
5
(𝑥 − 0)
𝑦1=
4
5
𝑥 & 𝑦2=-
4
5
𝑥
h , k
𝑎2
= 52
a=5
𝑏2 = 42
𝑏 = 4
𝑐2
= 𝑎2
+ 𝑏2
𝑐2
= 25 + 16 → 𝑐 = 41
Directrix: x=ℎ ±
𝑎
𝑒
e =
𝑐
𝑎
=
41
5
x=0 ±
5
41
5
𝑥1 = 3.9
𝑥2 = −3.9

29. Types of Conics Section
Graph:
𝑥2
52 −
𝑦2
42=1
𝑦2=-
4
5
𝑥 𝑦1=
4
5
𝑥
c
𝐹1𝐹2
𝐵1
𝐵2
𝑉1𝑉2
y
x

30. Thank you
for listening!!!