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1. PRE-CALCULUS
SIR JOSEPH

2. Specific objectives:
• illustrate the different types of conic sections: parabola, ellipse,
circle, hyperbola, and degenerate cases
• define a circle
• determine the standard form of equation of a circle
• graph a circle in a rectangular coordinate system

3. CONIC SECTIONS
A conic section, or simply conic, is a curve formed by the
intersection of a plane and a double right circular cone.

4. CONIC SECTIONS
A conic section, or simply conic, is a curve formed by the
intersection of a plane and a double right circular cone.

5. Circle
1
TYPESOF CONIC SECTIONS
Ellipse
2
Parabola
Hyperbola
3
4

6. TYPESOF CONIC SECTIONS
• Circle - the cutting plane is not parallel to any generator but is perpendicular to the axis

7. TYPESOF CONIC SECTIONS
• Ellipse- the cutting plane is not parallel to any generator

8. TYPESOF CONIC SECTIONS
• Parabola- the cutting plane is parallel to one and only one generator

9. TYPESOF CONIC SECTIONS
• Hyperbola- the cutting plane is parallel to the axis of the double cone
- is perpendicular to the bases of the two cones

10. TYPESOF CONIC SECTIONS
• Circle
• Ellipse
• Parabola
• Hyperbola

11. TYPESOF CONIC SECTIONS
• Circle
• Ellipse
• Parabola
• Hyperbola

12. Point
1
2
Line
3
Intersecting lines
DEGENERATE CASES:

13. DEGENERATECASES:
• Point - the cutting plane passes through the vertex of the cone

14. DEGENERATECASES:
• Line - the cutting plane passes through the generators

15. DEGENERATECASES:
• Intersecting lines - the cutting plane passes through the axis

16. DEGENERATECASES:
• Point
• Line
• Intersecting lines

17. DEGENERATECASES:
• Point
• Line
• Intersecting lines

18. KEY ELEMENTS:
• focus (F) - the fixed point of the conic
• directrix (d) - the fixed line d corresponding to the focus
• principal axis (a) - the line that passes through the focus and
perpendicular to the directrix
• vertec (V) - the point of intersection of the conic and its principal
• eccentricity (e) - the constant ratio

19. KEY ELEMENTS:

20. CIRCLE
A circle is a set of all coplanar points such that the distance
from a fixed point is constant. The fixed point is called the center of
the cicle and the constant distance form the center is called the
radius of the circle.

21. EQUATION OF A CIRCLE
Given that (h, k) is the center of the circle;
and (x, y) is a point in the circle
(x - h) 2 +(y - k)2
r
(x - h) 2 +(y - k)2
r
r2 = (x - h) 2 +(y - k)2
(x - h) 2 +(y - k)2 =r2
Standard form

22. EQUATION OF A CIRCLE
(x - h) 2 +(y - k)2 =r2
General form
x 2 +y 2 +Ax +By+C=0
Standard form

23. EQUATION OF A CIRCLE
To derive the equation of a circle whose center C is at the point (0, 0)
and with radius r, let P(x, y) be one of the points on the circle.
(x - 0) 2 +(y - 0)2
r
(x - 0) 2 +(y - 0)2
r
r2 = (x)2 +(y)2
r2 = x2 +y2

24. EXAMPLES:
Determine the standard form of equation of the circle given its
center and radius.
a. center (0, 0) , radius = 4
b. center (2, 5) , radius = 6
c. center (-2, 7) , radius = 4
d. center (-8, -5) , radius = 3

25. EXAMPLE a.:
center (0, 0) , radius =4
(x - h) 2 +(y - k)2 =r2

26. EXAMPLE A:
center (2, 5) , radius =6
(x - h) 2 +(y - k)2 =r2

27. EXAMPLE B:
center (2, 5) , radius =6
(x - h) 2 +(y - k)2 =r2 x 2 +y 2 +Ax +By+C=0

28. EXAMPLE C:
center (-2, 7) , radius =4
(x - h) 2 +(y - k)2 =r2

29. EXAMPLE D:
center (-8, -5) , radius =3
(x - h) 2 +(y - k)2 =r2

30. QUIZ:
Write the equation of the circle in general form and in standard
form.
1. center (3, -2) , radius =4
2. center (6, 5) , radius =8
3. center (0, 8), radius =1
1
(x - h) 2 +(y - k)2 =r2
x 2 +y 2 +Ax +By+C=0

31. THANK YOU
May GOD bless you always!