MTH 253 Calculus (Other Topics)Conic SectionsQuadratic Equation (2 variables)ParabolaParabola – Simple CasesParabola - ExamplesSlide 7Slide 8Slide 9Slide 10EllipseSlide 12Ellipse – Simple CasesEllipse - ExamplesSlide 15Slide 16HyperbolaSlide 18Hyperbola – Simple CasesSlide 20Hyperbola - ExamplesSlide 22Reflective Properties of ConicsShifting ConicsConics & Quadratic EquationsSlide 26MTH 253Calculus (Other Topics)Chapter 10 – Conic Sections and Polar CoordinatesSection 10.1 – Conic Sections and Quadratic EquationsCopyright © 2009 by Ron Wallace, all rights reserved.Conic SectionsThe intersection of a right circular cone and a plane will produce one of four curves:• Circle• Ellipse• Parabola• HyperbolaQuadratic Equation (2 variables)2 20Ax Bx y Cy Dx Ey F+ + + + + =It can, and will, be shown that any quadratic equation in two variables will be a conic section (and vice versa).• Begin with curves “centered” at the origin• Move the center• Rotate (B ≠ 0)ParabolaDefinitionThe set of all points equidistant from a fixed point (focus) and a line (directrix).The intersection of a right circular cone and a plane that is parallel to an element of the cone.directrixfocuspointparabolaParabola – Simple CasesFocus: (0, p)Directrix: y = –p 24x py=(0,p)(x,y)(x,–p)1 of 4Note that the points (2p,p) lie on this curve.Parabola - ExamplesGraph the following …24x y=212x y=212x y=Parabola – Simple CasesFocus: (0, –p)Directrix: y = p 24x py=-(0,–p)(x,y)(x,p)2 of 4Note that the points (2p,–p) lie on this curve.Parabola – Simple CasesFocus: (p, 0)Directrix: x = –p 24y px=(p,0)(x,y)(–p,y)3 of 4Note that the points (p,2p) lie on this curve.Parabola – Simple CasesFocus: (–p, 0)Directrix: x = p 24y px=-(–p,0)(x,y)(p,y)4 of 4Note that the points (–p,2p) lie on this curve.ParabolaThe point halfway between the focus and directrix is called the vertex.w/ the 4 simple cases, the originHow does the value of p affect the parabola?Small p  narrowLarge p  wide24x py=1p =14p =4p =EllipseDefinitionThe set of all points whose sum of distances to two fixed points (foci) is a constant.The intersection of a right circular cone and a plane that cuts clear through one of the nappes of the cone.EllipseTermsFocal axis: line containing the fociaka: major axisCenter: point halfway between the fociVertices: intersection of the ellipse with the focal axisEllipse – Simple CasesFoci: (c, 0)Vertices: (a, 0)Sum = 2a 2 22 22 2 21where x ya bb a c+ == -(a,0)(x,y)(–a,0)1 of 2(c,0)(–c,0)(0,b)(0,–b)Note that a > c & a > b.Ellipse - ExamplesGraph the following …2 2116 4x y+ =2 2112 2x y+ =Ellipse – Simple CasesFoci: (0, c)Vertices: (0, a)Sum = 2a 2 22 22 2 21where x yb ab a c+ == -(b,0)(x,y)(–b,0)2 of 2(0,c)(0,–c)(0,a)(0,–a)Note that a > c & a > b.EllipseHow does the difference between a & c affect the shape of the ellipse?a–c small  ovala–c large  roundWhat happens if c = 0?The foci and the center are all the same The result is a circle (b = a = r)The intersection of a cone with a plane that is perpendicular to the axis of the cone.2 22 2 22 21, where x yb a ca b+ = = -2 22 2 22 21 x yx y rr r+ = � + =HyperbolaDefinitionThe set of all points whose difference of distances to two fixed points (foci) is a constant.The intersection of a right circular cone and a plane that cuts intersects both nappes of the cone.HyperbolaTermsFocal axis: line containing the fociCenter: point halfway between the fociVertices: intersection of the hyperbola with the focal axisHyperbola – Simple CasesFoci: (c, 0)Vertices: (a, 0)Difference = 2a 2 22 22 2 21where x ya bb c a- == -(a,0)(x,y)(–a,0)1 of 2(c,0)(–c,0)Note that c > a & c > b.Hyperbola – Simple CasesAsymptotes? 2 22 22 2 21where x ya bb c a- == -(a,0)(x,y)(–a,0)1 of 2(c,0)(–c,0)(0,b)(0,–b)Note that c > a & c > b.by xa=�Hyperbola - ExamplesGraph the following …2 2116 4x y- =2 2112 20x y- =Hyperbola – Simple CasesFoci: (0, c)Vertices: (0, a)Difference = 2a 2 22 22 2 21where y xa bb c a- == -(a,0)(x,y)(–a,0)2 of 2(c,0)(–c,0)Note that c > a & c > b.(b,0)(–b,0)Asymptotes?ay xb=�Reflective Properties of ConicsParabola: Rays perpendicular to the directrix will reflect to the focus Ellipse: Rays emitting from one focus will reflect to the other focus.Hyperbola: Rays moving towards one focus will reflect to the other focus.See diagrams of applications on page 677.Shifting ConicsTo move the center of a conic to the point (h,k), replace x with x–h and replace y with y–k.2( ) 4 ( )x h p y k- = -2 22 2( ) ( )1x h y ka b- -- =2 22 2( ) ( )1x h y ka b- -+ =Conics & Quadratic EquationsComplete the squares and simplify.Parabola: A=0 or C=0Ellipse: A & C have the same signsHyperbola: A & C have opposite signs2 20Ax Cy Dx Ey F+ + + + =Conics & Quadratic EquationsExamples2 20Ax Cy Dx Ey F+ + + + =2 24 6 20 1 0x y x y+ + - + =23 6 20 5 0x x y+ - +