pre-cal first topic.pptx/PRE-CAL/PPTX.123
- 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.
- 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
- 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!