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# Calculus II - 19

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Stewart Calculus Section 10.5

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• ### Calculus II - 19

1. 1. 10.5 Conic SectionsThree types of conic sections: Parabola Ellipse Hyperbola
2. 2. A parabola is the set of points in a planethat are equidistant from a fixed point(called the focus) and a fixed line (calledthe directrix). The point half way between thefocus and the directrix is called the vertex.The line through the focus perpendicular tothe directrix is called the axis of theparabola. axis focus vertex directrix
3. 3. An equation of the parabola with focus ( , )and directrix = is = .An equation of the parabola with focus ( , )and directrix = is = . y = ( , ) x =
4. 4. An ellipse is the set of points in a plane thatthe sum of whose distances from two fixedpoints (called the foci) is a constant. Theline through the foci is called the major axisof the ellipse. The intersecting points of themajor axis and the ellipse are called thevertices. axis vertices foci
5. 5. The ellipse + = >has foci (± , ) where = , andvertices (± , ).The ellipse + = >has foci ( , ± ) where = , andvertices ( , ± ). y ( , ) + = ( , ) ( , ) ( , ) ( , ) x ( , )
6. 6. An hyperbola is the set of points in a planethat the difference of whose distances fromtwo fixed points (called the foci) is aconstant. The line through the foci is calledthe axis of the ellipse. The intersecting pointsof the major axis and the ellipse are calledthe vertices. Both branches of the hyperbolaapproach the asymptotes. foci axis vertices
7. 7. The hyperbola = has foci (± , )where = + , vertices (± , ) , andasymptotes = ±( / ) .The hyperbola = has foci ( ,± )where = + , vertices ( , ± ) , andasymptotes = ±( / ) . y ( , ) ( , ) x
8. 8. Ex: What can you say about the conic? + + =
9. 9. Ex: What can you say about the conic? + + =Complete the square: ( ) ( )+ =
10. 10. Ex: What can you say about the conic? + + =Complete the square: ( ) ( )+ = ( + ) ( + )+ =
11. 11. Ex: What can you say about the conic? + + =Complete the square: ( ) ( )+ = ( + ) ( + )+ = ( ) ( ) + =
12. 12. Ex: What can you say about the conic? + + =Complete the square: ( ) ( )+ = ( + ) ( + )+ = ( ) ( ) + = ( ) ( ) =
13. 13. Ex: What can you say about the conic? + + =Complete the square: ( ) ( )+ = ( + ) ( + )+ = ( ) ( ) + = ( ) ( ) =It is a hyperbola, = , = , = + = .The foci are ( , ± ). The vertices are ( , )and ( , ). The asymptotes are − = ± ( − ).