Conic Sections_Circle.pptx

LL II FF E E PP EERR FF O ORRM MA AN NCCE E OOU U TTC COOM ME E Mindful, Self-directed LEARNER S and ROLE MODELS I am a mindful, self-directed learner and role model, consciously expressing my Faith. EESSSSE ENN TT II A...

LL II FF E E PP EERR FF O ORRM MA AN NCCE E OOU U TTC COOM ME E Mindful, Self-directed LEARNER S and ROLE MODELS I am a mindful, self-directed learner and role model, consciously expressing my Faith. EESSSSE ENN TT II A A LL PP E ERR FF O OR RM MA AN NC CEE O OUU TTC COOM MEE Assess and strengthen their own mathematical skills for continued learning and personal fulfillment. DEGENERAT E OR NONDEGENE RATE ACTIVITY ACTIVITY Hyperbola NONDEGENERATE Line DEGENERATE Point DEGENERATE Circle NONDEGENERATE Ellipse NONDEGENERATE Conic secti ons The The conic conic sections sections The Greek mathematicians were developed the geometric properties of parabolas, circles, ellipses, and hyperbolas. These curves are called conics because they are formed through by passing a plane through right circular cones with two nappes. The The conic conic sections sections Generator or Element of the cone is the line lying in the cone and all generators of a cone contain the vertex V. The The conic conic sections sections Axis of the cone is a line from the vertex to the center of the base of a cone. The The conic conic sections sections If the cutting plane is parallel to the generator then we have a parabola. The The conic conic sections sections If the cutting plane is parallel to the two generators then we have a hyperbola. The The conic conic sections sections If the cutting plane is not parallel to any generator then we have an ellipse. The The conic conic sections sections Circle is formed when the cutting plane which intersects each generator is perpendicular to the axis of the cone. The The conic conic sections sections Circle is a special kind of ellipse. Nondegenerate Nondegenerate conics conics Nondegenerate Nondegenerate conics conics A conic is nondegenerate if the plane does not pass through the vertex of the cone. Degenerate Degenerate conics conics Intersecting Line Point Lines Degenerate Degenerate conics conics A parabola degenerates into a line if the plane contains the vertex and only one generator. Line Degenerate Degenerate conics conics An ellipse degenerates into a point if the plane contains the vertex of the cone but does not contain a generator. Point Degenerate Degenerate conics conics A hyperbola degenerates into two intersecting lines if the plane contains the vertex and two generators. Intersecting Lines degenerate degenerate conics conics A conic is degenerate if the plane pass through the vertex of the cone. Quadratic Quadratic equation equation Quadratic Equation A general equation of the second degree in and. 2 2 𝐴𝑥 + 𝐵𝑥𝑦 +𝐶 𝑦 + 𝐷𝑥+ 𝐸𝑦 + 𝐹 =0 Where and with real coefficients and at least one of and is non-zero  Quadratic Quadratic equation equation The graphs of this equation will depend on the sign of and the value of discriminant  Quadratic Quadratic equation equation Nondegenerate Conics Degenerate Conics Discriminant ≠ 0 Discriminant = 0 If and , the graph is a circle Point If , the graph is parabola One line If , the graph is an ellipse Point If , the graph is a hyperbola Two intersecting line EXAMPLE EXAMPLE 1 1 2 2 𝑥 + 𝑦 − 2 𝑥 − 4 𝑦 + 4= 0 the conic is non degenerate and the conic is ellipse Try Try this! this! 2 2 𝑥 − 2 𝑥𝑦 − 8 𝑦 +4 𝑥+ 2 𝑦 +3=0 2 2 𝐴𝑥 + 𝐵𝑥𝑦 +𝐶 𝑦 + 𝐷𝑥+ 𝐸𝑦 + 𝐹 =0 2 2 2 𝑑= [ 𝐹 ( 4 𝐴𝐶 − 𝐵 ) + 𝐵𝐷𝐸 − 𝐴 𝐸 − 𝐶 𝐷 ] ,, , Try Try this! this! 2 2 𝑥 − 2 𝑥𝑦 − 8 𝑦 +4 𝑥+ 2 𝑦 +3=0 2 2 2 𝑑= [ 𝐹 ( 4 𝐴𝐶 − 𝐵 ) + 𝐵𝐷𝐸 − 𝐴 𝐸 − 𝐶 𝐷 ] 𝑑= ¿ 𝑑= − 108 − 16 − 4 +12 𝑑= 0 Degenerate Try Try this! this! 2 2 𝑥 − 2 𝑥𝑦 − 8 𝑦 +4 𝑥+ 2 𝑦 +3=0 2 𝐵 − 4 𝐴𝐶 2 2 𝐵 − 4 𝐴𝐶=( − 2 ) − 4 ( 1 ) ( − 8 ) =36 Since 36 is positive and , therefore, the conic degenerate into two intersecting lines. Try Try this! this! 2 2 𝑥 − 2 𝑥𝑦 − 8 𝑦 +4 𝑥+ 2 𝑦 +3=0 The conic is degenerate and the conic degenerate into two intersecting lines. Try Try this! this! 2 2 2 𝑥 +6 𝑥𝑦 +3 𝑦 + 4 𝑥+6 𝑦=0 2 2 𝐴𝑥 + 𝐵𝑥𝑦 +𝐶 𝑦 + 𝐷𝑥+ 𝐸𝑦 + 𝐹 =0 2 2 2 𝑑= [ 𝐹 ( 4 𝐴𝐶 − 𝐵 ) + 𝐵𝐷𝐸 − 𝐴 𝐸 − 𝐶 𝐷 ] ,, , Try Try this! this! 2 2 2 𝑥 +6 𝑥𝑦 +3 𝑦 + 4 𝑥+6 𝑦=0 2 2 2 𝑑= [ 𝐹 ( 4 𝐴𝐶 − 𝐵 ) + 𝐵𝐷𝐸 − 𝐴 𝐸 − 𝐶 𝐷 ] 𝑑= ¿ 𝑑= 0 +144 − 72 − 4 𝑑=24 Nondegenerate Try Try this! this! 2 2 2 𝑥 +6 𝑥𝑦 +3 𝑦 + 4 𝑥+6 𝑦=0 2 𝐵 − 4 𝐴𝐶 2 2 𝐵 − 4 𝐴𝐶 =( 6 ) − 4 ( 2 ) ( 3 )= 12 Since 12 is positive and , therefore, the conic is hyperbola. Try Try this! this! 2 2 2 𝑥 +6 𝑥𝑦 +3 𝑦 + 4 𝑥+6 𝑦=0 The conic is nondegenerate and the conic is hyperbola Try Try this! this! 2 2 3 𝑥 +5 𝑥𝑦 +2 𝑦 − 7 𝑥+ 5 𝑦= 0 2 2 𝐴𝑥 + 𝐵𝑥𝑦 +𝐶 𝑦 + 𝐷𝑥+ 𝐸𝑦 + 𝐹 =0 2 2 2 𝑑= [ 𝐹 ( 4 𝐴𝐶 − 𝐵 ) + 𝐵𝐷𝐸 − 𝐴 𝐸 − 𝐶 𝐷 ] ,, , Try Try this! this! 2 2 3 𝑥 +5 𝑥𝑦 +2 𝑦 − 7 𝑥+ 5 𝑦= 0 2 2 2 𝑑= [ 𝐹 ( 4 𝐴𝐶 − 𝐵 ) + 𝐵𝐷𝐸 − 𝐴 𝐸 − 𝐶 𝐷 ] 𝑑= ¿ 𝑑= 0 − 175 − 75 − 9 𝑑=−348 Nondegenerate Try Try this! this! 2 2 3 𝑥 +5 𝑥𝑦 +2 𝑦 − 7 𝑥+ 5 𝑦= 0 2 𝐵 − 4 𝐴𝐶 2 2 𝐵 − 4 𝐴𝐶 = ( 5 ) − 4 ( 3 ) ( 2 )= 1 Since 1 is positive and , therefore, the conic is hyperbola. Try Try this! this! 2 2 3 𝑥 +5 𝑥𝑦 +2 𝑦 − 7 𝑥+ 5 𝑦= 0 The conic is nondegenerate and the conic is hyperbola activity activity circle circle  circle circle A circle is a set of all points in a plane diu s equidistant from a fixed point. The fixed point is Ra called the center of the circle, and the constant equal distance is called the radius. This is a special kind of ellipse. Center Parts Parts of of a a circle circle The center, denoted by The radius, denoted by The diameter, , a line segment joining two 𝑟 points on the circle and passing through the center and whose length is 𝐶 𝑑 circle circle 𝐶𝑃 =𝑟 2 2 √ ( 𝑥 − h) +( 𝑦 − 𝑘 ) =𝑟 𝑟 2 2 ( 𝑥 − h ) + ( 𝑦 − 𝑘 ) =𝑟 2 Standard Form ( 𝑥 2 − 2 h 𝑥 +h2 ) + ( 𝑦 2 − 2 𝑘𝑦 + 𝑘2 )=𝑟 2 𝑥2 + 𝑦 2 −2 h𝑥 −2𝑘𝑦+ ( h2 +𝑘2 −𝑟 2 )=0 General Form STANDARD STANDARD EQUATION EQUATION STANDARD EQUATION OF A CIRCLE The standard equation of a circle with center at and radius is given by Where GENERAL GENERAL EQUATION EQUATION GENERAL EQUATION OF A CIRCLE The general equation of a circle is 2 2 𝐴𝑥 + 𝐵 𝑦 + 𝐶𝑥 + 𝐷𝑦 + 𝐸=0 Where A and B are equal; and EXAMPLE EXAMPLE 1 1 Find the center and radius of the 2 2 circle 𝑥 + 𝑦 −4 𝑥 −10 𝑦 + 13=0 2 2 𝐴𝑥 + 𝐵 𝑦 + 𝐶𝑥 + 𝐷𝑦 + 𝐸=0 General Form EXAMPLE EXAMPLE 1 1 2 2 𝑥 + 𝑦 −4 𝑥 −10 𝑦 +13=0 2 2 𝑥 + 𝑦 −4 𝑥 −10 𝑦 +13−13=0 −13 2 2 𝑥 + 𝑦 −4 𝑥 −10 𝑦 =−13 2 2 Last Term: ( 𝑥 − 4 𝑥 ) + ( 𝑦 −10 𝑦 ) =−13 2 2 ( 𝑥 − 4 𝑥 +4 ) + ( 𝑦 − 10 𝑦 +25 ) =− 13 +4+25 2 2 ( 𝑥 − 2 ) + ( 𝑦 − 5 ) =16 EXAMPLE EXAMPLE 1 1 2 2 2 2 2 ( 𝑥 − 2 ) + ( 𝑦 − 5 ) =16 ( 𝑥 − h ) + ( 𝑦 − 𝑘 ) =𝑟 2 2 √ ( 𝑥 − 2 ) + ( 𝑦 − 5 ) =√ 16 The center of the circle is at and the ( 𝑥 − 2 ) + ( 𝑦 − 5 ) =4 radius is 4. EXAMPLE EXAMPLE 1 1 Find the equation of the circle The center of the circle is at and the radius is 4. 2 2 𝑥 + 𝑦 −4 𝑥 −10 𝑦 +13=0 GRAPHING GRAPHING CIRCLE CIRCLE = 4 The center of the circle 𝑟 is at and the radius is 4.  2 2 𝑥 + 𝑦 −4 𝑥 −10 𝑦 +13=0 The circle of the center of the circle is at and the radius is 4. EXAMPLE EXAMPLE 2 2 Find the equation of the circle The center of the circle is at and the radius is 1. EXAMPLE EXAMPLE 2 2 The center of the circle is at and the radius is 1. GRAPHING GRAPHING CIRCLE CIRCLE = 1 The center of the circle 𝑟 is at and the radius is 1. Try Try this! this! Find the equation of the circle The center of the circle is at and the radius is 12. 𝟐 𝟐 𝒙 + 𝒚 − 𝟖 𝒙 − 𝟏𝟔 𝒚 − 𝟔𝟒=𝟎 Try Try this! this! Find the center and radius of the 2 2 circle 𝑥 + 𝑦 − 6 𝑥 − 8 𝑦 +10=0 The center of the circle is at and the radius is. Try Try this! this! Find the equation of the circle The center of the circle is at and the radius is 8. 𝟐 𝟐 𝒙 + 𝒚 − 𝟏𝟒 𝒙 − 𝟏𝟖 𝒚 − 𝟔𝟔=𝟎 Seatwork Seatwork 1 1 1. Find the center and radius of the following circles. a. b. c. Seatwork Seatwork 1 1 2. In each of the following, find the equation of the circle determined by the given conditions, and draw the figure. a. Radius 13, center at b. Radius 5, center at  activity activity Column A Column B b _______1. What is the standard form of a. Ellipse the equation of a circle? b. d _______2. What type of conic section is a set of all points in a plane equidistant c. from a fixed point? d. Circle c _______3. What is the general form of the equation of a circle? Than k you!