PRE-CALCULUS (Lesson 1-Conic Sections and Circles).pptx

introduction of Conic Section (Circles, Ellipse, Parabola, Hyperbola)

1. PRE-CALCULUS Ma’am Grace

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 TYPES OF CONIC SECTIONS Ellipse 2 Parabola 3 Hyperbola 4

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

7. TYPES OF CONIC SECTIONS Ellipse - the cutting plane is not parallel to any generator

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

9. TYPES OF CONIC SECTIONS Hyperbola - the cutting plane is parallel to the axis of the double cone - is perpendicular to the bases of the two c ones

10. TYPES OF CONIC SECTIONS Circle Ellipse Parabola Hyperbola

11. TYPES OF CONIC SECTIONS Circle Ellipse Parabola Hyperbola

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

13. DEGENERATE CASES: Point - the cutting plane passes through the vertex of the cone

14. DEGENERATE CASES: Line - the cutting plane passes through the generators

15. DEGENERATE CASES: Intersecting lines - the cutting plane passes through the axis

16. DEGENERATE CASES: Point Line Intersecting lines

17. DEGENERATE CASES: 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 r 2 = (x - h ) 2 + ( y - k ) 2 ( x - h ) 2 + ( y - k ) 2 = r 2 Standard form

22. EQUATION OF A CIRCLE ( x - h ) 2 + ( y - k ) 2 = r 2 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 - ) 2 + (y - ) 2 r (x - ) 2 + (y - ) 2 r r 2 = (x) 2 + (y) 2 r 2 = x 2 + y 2

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

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

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

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

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

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

30. QUIZ: Write the equation of the circle in general form and in standard form . center (3, -2) , radius = 4 center (6, 5) , radius = 8 center (0, 8), radius = 11 ( x - h ) 2 + ( y - k ) 2 = r 2 x 2 + y 2 + Ax + By + C = 0

31. THANK YOU May GOD bless you always!