SIBY SEBASTIAN PGT (MATHS) KENDRIYA VIDYALAYA GILL NAGAR

1. CONIC SECTION SIBY SEBASTIAN PGT (MATHS) KENDRIYA VIDYALAYA GILL NAGAR

2. FORMATION OF CONICS Let l be a fixed line and m be a another line intersecting l at V and inclination to it at an angle α.

3. FORMATION OF CONICS • If ‘m’ rotates around ‘l’ such that ‘α’ remain constant then surface generated is a double-napped right circular hollow cone. ‘V’ is called vertex ‘l’ is called axis and ‘m’ is generator of the cone.

4. CONIC SECTIONS Conics or Conic Sections are defined as the intersection of a plane and a cone .

8. CONIC SECTION DEFINITION

9. CIRCLE A circle is a set of all points in a plane that are equidistant from a fixed point in the plane

10. PARABOLA A parabola is the set of all points in a plane that are equidistant from a fixed line (called the directrix) and a fixed point ( called the focus)

11. ELLIPSE An ellipse is a set of all points in a plane ,the sum of whose distances from two fixed points in a plane is constant.

12. HYPERBOLA • The set of all points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.

13. PARABOLA A parabola is the set of all points in a plane such that each point in the set is equidistant from a line called the directrix and a fixed point is called the focus.

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

15. y l Let P(x,y) be any point on the Parabola. PB = x+a Directrix By def PF = PB PF2 = PB2 P(x,y) B PF2 = (x-a)2 + y2 So, (x-a)2 + y2 = ( x+a)2  x F(a,0) On Simplifying we get y2 =4ax Where a  0. O(0,0) x+a =0 Y’

16. Other forms of Parabola

17. Other forms of Parabola

18. Other forms of Parabola

19. OTHER FORMS OF PARABOLA y2=4ax y2=-4ax

20. PARABOLA

21. GENERAL EQUATION OF A PARABOLA The General Equation of a PARABOLA with vertex (h ,k ) is given by (x –h)2 = 4a(y – k) and (y - k)2 = 4a(x – h)2

22. ELLIPSE The plane can intersect one nappe of the cone at an angle to the axis resulting in an ellipse.

23. ELLIPSE An ellipse is a set of all points in a plane ,the sum of whose distances from two fixed points in a plane is constant.

24. major axis vertex latus rectum minor axis length of major axis = 2a length of the minor axis =2b length of latus rectum = 2b2/a

25. RELATION BETWEEN a, b, c F1P + F2P = F1O + OP + F2P = c + a + a – c = 2a F1Q + F2Q = + 2a = 2 a2 = b2 + c2 Q(0,b) o O P(a,o) R(-a,o) F2(C,0) F1(-C,0) S(0,-b)

26. ECCENTRICITY • ECCENTRICITY OF AN ELLIPSE IS DEFINED AS THE RATIO OF THE DISTANCE FROM THE CENTRE OF THE ELLIPSE TO ONE OF THE FOCI AND TO ONE OF THE VERTICES. e = e < 1

27. PF1 + PF2 = A1F2+AF1 PF1 + PF2 = (c + a)+(a-c) PF1 + PF2 = 2a

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

29. standard equation of an ellipse

30. Other form of Ellipse

31. LATUS RECTUM • o B C D LATUS RECTUM OF AN ELLIPSE IS A LINE SEGMENT PERPENDICULAR TO THE MAJOR AXIS THROUGH ANY OF THE FOCI AND WHOSE END POINTS LIE ON THE ELLIPSE l O B(a,O) A(-a,O) F1(-C,O) F2(C,O) C A D

32. LATUS RECTUM • o Equation of the ellipse is B C D ( ae , l) l O B(a,O) A(-a,O) F1(-C,O) F2(C,O) C A D