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PRE-CALCULUS (Lesson 1-Conic Sections and Circles).pptx
- 1.
- 2. Specific objectives: • illustratethe 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 conicsection, or simply conic, is a curve formed by the intersection of a plane and a double right circular cone.
- 4. CONIC SECTIONS A conicsection, or simply conic, is a curve formed by the intersection of a plane and a double right circular cone.
- 5. Circle 1 TYPES OF CONICSECTIONS Ellipse 2 Parabola 3 Hyperbola 4
- 6. TYPES OF CONICSECTIONS • Circle - the cutting plane is not parallel to any generator but is perpendicular to the axis
- 7. TYPES OF CONICSECTIONS • Ellipse - the cutting plane is not parallel to any generator
- 8. TYPES OF CONICSECTIONS • Parabola - the cutting plane is parallel to one and only one generator
- 9. TYPES OF CONICSECTIONS • Hyperbola- the cutting plane is parallel to the axis of the double cone - is perpendicular to the bases of the two cones
- 10. TYPES OF CONICSECTIONS • Circle • Ellipse • Parabola • Hyperbola
- 11. TYPES OF CONICSECTIONS • Circle • Ellipse • Parabola • Hyperbola
- 12.
- 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: • Intersectinglines - 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.
- 20. CIRCLE A circle isa 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 ACIRCLE 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 ACIRCLE (x - h) 2 + (y - k)2 = r2 General form x 2 + y 2 + Ax + By + C = 0 Standard form
- 23. EQUATION OF ACIRCLE 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 standardform 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 equationof 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 = 11 (x - h) 2 + (y - k)2 = r2 x 2 + y 2 + Ax + By + C = 0
- 31. THANK YOU May GODbless you always!