1 / 38

Conics

Conics. Conic Sections. (1) Circle. A circle is formed when i.e. when the plane  is perpendicular to the axis of the cones. Conic Sections. (2) Ellipse. An ellipse is formed when

ataret
Download Presentation

Conics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Conics

  2. Conic Sections (1) Circle A circle is formed when i.e. when the plane  is perpendicular to the axis of the cones.

  3. Conic Sections (2) Ellipse An ellipse is formed when i.e. when the plane  cuts only one of the cones, but is neither perpendicular to the axis nor parallel to the a generator.

  4. Conic Sections (3) Parabola A parabola is formed when i.e. when the plane  is parallel to a generator.

  5. Conic Sections (4) Hyperbola A hyperbola is formed when i.e. when the plane  cuts both the cones, but does not pass through the common vertex.

  6. y P(x,y) M(-a,0) O focus F(a,0) x Parabola A parabola is the locus of a variable point on a plane so that its distance from a fixed point (the focus) is equal to its distance from a fixed line (the directrix x = - a).

  7. Form the definition of parabola, PF = PN standard equation of a parabola

  8. axis of symmetry vertex latus rectum (LL’) mid-point of FM = the origin (O) = vertex length of the latus rectum = LL’= 4a

  9. Other forms of Parabola

  10. Other forms of Parabola

  11. Other forms of Parabola

  12. Ellipses An ellipse is the locus of a variable point on a plane so that the sum of its distance from two fixed points is a constant. P’(x,y) P’’(x,y)

  13. Let PF1+PF2 = 2a where a > 0

  14. standard equation of an ellipse

  15. major axis = 2a vertex lactus rectum minor axis = 2b length of semi-major axis = a length of the semi-minor axis = b length of lactus rectum =

  16. Other form of Ellipse

  17. Hyperbolas A hyperbola is the locus of a variable point on a plane such that the difference of its distance from two fixed points is a constant. P’(x,y)

  18. Let |PF1-PF2|= 2a where a > 0

  19. standard equation of a hyperbola

  20. transverse axis vertex lactus rectum conjugate axis length of lactus rectum = length of the semi-transverse axis = a length of the semi-conjugate axis = b

  21. asymptote equation of asymptote :

  22. Other form of Hyperbola :

  23. Rectangular Hyperbola If b = a, then The hyperbola is said to be rectangular hyperbola.

  24. equation of asymptote :

  25. If the rectangular hyperbola x2 – y2 = a2 is rotated through 45o about the origin, then the coordinate axes will become the asymptotes. equation becomes :

  26. Simple Parametric Equations and Locus Problems x = f(t) y = g(t) parametric equations parameter Combine the two parametric equations into one equation which is independent of t. Then sketch the locus of the equation.

  27. Equation of Tangents to Conics general equation of conics : • Steps : • Differentiate the implicit equation to find . • Put the given contact point (x1,y1) into • to find out the slope of tangent at that point. • (3) Find the equation of the tangent at that point.

  28. OR

More Related