Maths Project On Conic Sections.pptx

Cones construction

1. MATHS PROJECT XI A

2. Conic Sections

3. history The names parabola and hyperbola are given by APOLLONIUS .These curves are in fact, known as CONIC SECTIONS or more commonly CONICS because they can be obtained as intersections of a plane with a double napped right circular cone. These curves have a very wide range of application in fields such as planetary motion, design of telescopes and reflectors in flash lights and automobile headlights, etc .

4. Sections of a Cone Let l be a fixed vertical line and m be another line intersecting it at a fixed point V and inclined to it at an angle α. V  l m Suppose we rotate the line m around the line l in such a way that the angle α remains constant. Then the surface generated is a double-napped right circular hollow cone herein after referred as cone and extending indefinitely far in both directions.

5. The point V is called the vertex; The line l is the axis of the cone. The rotating line m is called a generator of the cone. The vertex separates the cone into two parts called nappes. If we take the intersection of a plane with a cone, the section so obtained is called a conic section. Thus, conic sections are the curves obtained by intersecting a right circular cone by a plane. Let ß be the angle made by the intersecting plane with the vertical axis. The intersection of the plane can take place in any axis.

6. CONIC SECTION When a right circular cone is intersected by a plane, the curves obtained are known as conic sections. If the plane which cuts the cone is parallel the generator, then the conic section obtained is called PARABOLA. When the plane which cuts the cone is not parallel to the generator, then the conic section obtained is called an ELLIPSE. When the plane which cuts the cone is parallel to the axis, then the conic section is called HYPERBOLA .

7. Conic Sections When the plane cuts the nappe (other than the vertex) of the cone, we have the following situations : (a) When β = 90º, the section is a circle. (b) When α < β < 90º, the section is an ellipse.

8. (c) When β = α; the section is a parabola (d) When 0 ≤ β < α; the plane cuts through both the nappes and the curves of intersection is a hyperbola

9. When the plane cuts at the vertex of the cone, we have the following different cases: (a) When α < β ≤ 90o, then the section is a point (b) When β = α, the plane contains a generator of the cone and the section is a straight line. It is the degenerated case of a parabola.

10. Circle

11. C i r c l e The fixed point is called the centre of the circle and the distance from the centre to a point on the circle is called the radius of the circle A Circle is the set of all points in a plane that are equidistant from a fixed point in the plane.

12. Given C ( h, k) be the centre and r the radius of circle. Let P(x, y) be any point on the circle. Then, by the definition, | CP | = r . By the distance formula, (x-h) 2 + (y-k) 2 = r 2 EQUATION OF A CIRCLE

13. PARABOLA

14. P a r a b o l a A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane. The fixed line is called the directrix of the parabola and the fixed point F is called the focus .(‘Para’ means ‘for’ and ‘bola’ means ‘throwing’, i.e., the shape described when you throw a ball in the air). A line through the focus and perpendicular to the directrix is called the axis of the parabola. The point of intersection of parabola with the axis is called the vertex of the parabola .

15. Standard Equation of Parabola Y O Y’ X X’ F (a,0) M l Let F be the focus and l the directrix. Let FM be perpendicular to the directrix and bisect FM at the point O. Produce MO to X. By the definition of parabola, the mid-point O is on the parabola and is called the vertex of the parabola. Take O as origin, OX as x-axis and OY perpendicular to it as the y-axis. Let the distance from the directrix to the focus be 2 a.

16. Let P( x, y) be any point on the parabola such that PF = PB, ... (1) where PB is perpendicular to l. The coordinates of B are (– a, y). By the distance formula, we have PF = ( x – a) 2 + y 2 and PB = (x + a) 2 Since PF = PB, we have ( x – a) 2 + y 2 = (x + a) 2 x 2 – 2ax + a 2 + y 2 = x 2 + 2ax + a 2 y 2 = 4ax ( a > 0). …(2) Then, the coordinates of the focus are (a, 0), and the equation of the directrix is x + a = 0. Y O Y’ X X’ F (a,0) x = - a M l P (x,y) (-a, y) B Or, And so, P(x, y) lies on the parabola.

17. Latus Rectum Y O Y X X’ F (a, 0) M l Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola. To find the Length of the latus rectum of the parabola y2 = 4ax. By the definition of the parabola, AF = AC. But AC = FM = 2a Hence AF = 2a. And since the parabola is symmetric with respect to x-axis AF = FB and so AB = Length of the latus rectum = 4a. C A B

18. PARABOLA OF THE TYPE, y 2 =-4ax , a>0 CHARACTRISTICS Focus (-a,0) Equation of the axis is y= 0 Equation of the directrix is x-a= 0 Length of the latus rectum = 4a units

19. PARABOLA OF THE TYPE , x 2 =4ay , a>0 CHARACTRISTICS Focus (0,a) Equation of the axis is x= 0 Equation of the directrix is y+a = 0 Length of the latus rectum = 4a units

20. PARABOLA OF THE TYPE , x 2 =-4ay , a>0 CHARACTRISTICS Focus (0,-a) Equation of the axis is x= 0 Equation of the directrix is y-a= 0 Length of the latus rectum = 4a units

21. ellipse

22. Ellipse An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant. The two fixed points are called the foci (plural of ‘ focus’) of the ellipse. P 1 P 2 P 3 Focus F 1 Focus F 2 P 1 F 1 + P 1 F 2 = P 2 F 1 + P 2 F 2 = P 3 F 1 + P 3 F 2

23. The mid point of the line segment joining the foci is called the centre of the ellipse. The line segment through the foci of the ellipse is called the major axis. The line segment through the centre and perpendicular to the major axis is called the minor axis. The end points of the major axis are called the vertices of the ellipse. We denote the length of the major axis by 2 a, the length of the minor axis by 2b and the distance between the foci by 2 ae (2c). Thus the length of the semi major axis is a and that of the semi minor axis is b.

24. STANDARD EQUATION OF THE ELLIPSE:-

25. Let F1 and F2 be the foci and 0 be the mid-point of the line segment F1 , F2. let 0 be the origin and the line from 0 through F1 be the positive s-axis and through F2 as the negative axis. Let the line through 0 perpendicular to the x-axis be the y-axis. Let the coordinates of F1 and F2 be F1(ae,0) and F2(-ae,0). Let P( x,y ) be any point on the ellipse. Such that sum of the distance from P to the two foci be 2a. By definition of the ellipse, PF1+PF2= constant PF1+PF2=2a

26. = 2a Squaring both the sides, x 2 +2xae+a 2 e 2 +y 2 =4a 2 -4a +y 2 +(x- ae ) 2 +y 2 x 2 +2xae+a 2 e 2 =4a 2 -4a +x 2 -2xae+a 2 e 2 4xae=4a 2 -4a 4a( xe )= 4a [a- ] =a- xe Squaring again, x 2 -2xae+a 2 e 2 +y 2 = a 2 -2xae+x 2 e 2 (x 2 -x 2 e 2 )+y 2 = a 2 -a 2 e 2

27. x 2 (1-e 2 )+y 2 = a 2 (1-e) 2 , where b 2 = a 2 (1- e 2 )

28. Here AB and CD are the latus rectum Let AF1= k(say) Therefore, ( ae,k ) Since A( ae,k ) lies on , b 2 =a 2 (1-e 2 )

29. Therefore, length of the latus rectum = units ECCENTRICITY:- The eccentricity of an ellipse is the ratio of the distance from the center of the ellipse to one of the foci and to one of the vertices of the ellipse and is denoted by ‘e’.

30. Ellipse of the type, ( Major axis along y-axis ) Vertex (0,±a) Foci (0, ± ae ) Length of the minor axis =2a Length of the latus rectum = units

31. hyperbola

32. The Hyperbola The plane that intersects the cone is parallel to the axis of symmetry of the cone.

33. HYPERBOLA:- A hyperbola is the set of all the points in a plane the difference of whose distance from two fixed points in the plane is the constant.

34. The two fixed points are called the foci of the hyperbola. The mid-point of the line segment joining the foci is called the centre of the hyperbola. The line through the foci is called the TRANSVERSE AXIS and the line through the centre and perpendicular to the transverse axis is called the CONJUGATE AXIS. The points at which the hyperbola intersects the transverse axis are called the VERTEX of the hyperbola. Distance between two foci =2ae. Length of the transverse axis =2a and the length of the conjugate axis =2b.

35. STANDARD EQUATION OF HYPERBOLA:-

36. Let F1 and F2 be the foci 0 be the mid-point of the line segment F1 ,F2. Let 0 be the origin and the line through 0, through F1 be the positive x-axis and through F2 be the negative x-axis. The line through 0 perpendicular to x-axis be the y-axis. Let the coordinates of F1 and F2 be F1( ae , 0) and F2(- ae , 0). Let P( x,y ) be any point on the hyperbola, such that PF 2 -PF 1 =2a PF 2 =2a+PF 1 ( x+ae ) 2 +y 2 =4a 2 +4a + (x- ae ) 2 +y 2

37. x 2 +2xae+a 2 e 2 =4a 2 +4a +x 2 -2xae+a 2 e 2 4xae-4a 2 =4a 4a( xe -a) =4a ( xe -a) 2 =(x- ae ) 2 +y 2 x 2 e 2 -2xae+a 2 =x 2 -2xae+a 2 e 2 +y 2 (x 2 e 2 -x 2 )-y 2 =a 2 e 2 -a 2 x 2 (e 2 -1)-y 2 =a 2 (e 2 -1) , where b 2 =a 2 (e 2 -1)

38. HYPERBOLA OF THE TYPE Vertices (0, ±a) Foci (0, ± ae )

39. Thank You 39 Thank you for your kind attention. This power point presentation was created by Divya of class XI A , K.V A.S.C Centre[S].