Clss-XI Mthemtics Conic Sections Chpter-11 Chpter Notes Key Concepts 1. Let be fixed verticl line nd m be nother line intersecting it t fixed point V nd inclined to it t nd ngle On rotting the line m round the line in such wy tht the ngle remins constnt surfce is generted is double-npped right circulr hollow cone. 2. The point V is clled the vertex; the line is the xis of the cone. The rotting line m is clled genertor of the cone. The vertex seprtes the
cone into two prts clled nppes. 3. The sections obtined by cutting double npped cone with plne re clled conic sections. Conic sections re of two types (i) degenerte conics (ii) non degenerte conics. 4. If the cone is cut t its vertex by the plne then degenerte conics re obtined. 5. If the cone is cut t the nppes by the plne then non degenerte conics re obtined. 6. Degenerte conics re point, line nd double lines. 7. Circle, prbol, ellipse nd hyperbol re degenerte conics. 8. When the plne cuts the nppes (other thn the vertex) of the cone, degenerte conics re obtined.
() When = 90 o, the section is circle. The plne cuts the cone horizontlly. (b) When < < 90 o, the section is n ellipse. Ellipse The plne cuts one prt of the cone in n inclined mnner
(c) When = ; the section is prbol. Prbol The plne cuts the cone in such wy tht it is prllel to genertor (d) When 0 < ; the plne cuts through both the nppes the curve of intersection is hyperbol. Hyperbol The plne cuts both prts of the cone. 9. When the plne cuts t the vertex of the cone, we hve the following different cses:
() When < 90 0, then the section is point. Point degenerted cse of circle. (b) When =, the plne contins genertor of the cone nd the section is stright line. Line It is the degenerted cse of prbol.
(c) When 0 <, the section is pir of intersecting stright lines. It is the degenerted cse of hyperbol. Double Line 10. A circle is the set of ll points in plne tht re equidistnt from fixed point in the plne. 11. The fixed point is clled the centre of the circle nd the distnce from the centre to point on the circle is clled the rdius of the circle. In the circle O is the centre nd O A = OB = OC re the rdii. 12. If the centre of circle is (h, k) nd the rdius is r, then the eqution of the circle is given by (x - h) 2 + (y - k) 2 = r 2 13. A circle with rdius of length zero is point circle. 14. If the centre of circle is t origin nd the rdius is r, then the eqution of the circle is given by x 2 + y 2 = r 2 15. A prbol is the locus of point, which moves in plne in such wy tht its distnce from fixed point (not on the line) in the plne is equl to
its distnce from fixed stright line in the sme plne. 15. If the fixed point is on the fixed line then the set of points which re equidistnt from the line nd focus will be stright line which psses through the fixed point focus nd perpendiculr to the given line. This stright line is the degenerte cse of the prbol. 16. The fixed line is clled the directrix of the prbol nd the fixed point F is clled the focus. 17. Pr mens for nd bol mens throwing. The pth tken by the trjectory of rocket rtillery etc re prbolic. One of nture's best known pproximtions to prbols is the pth tken by body projected upwrd nd obliquely to the pull of grvity, s in the prbolic trjectory of golf bll. 18. A line through the focus nd perpendiculr to the directrix is clled the xis of the prbol. The point of intersection of prbol
with the xis is clled the vertex of the prbol. 19. A chord of prbol is the line segment joining ny two points on the prbol. If the chord psses through the focus it is focl chord. LM nd PQ re both chords but PQ is focl chord. 20. The chord which psses through the focus is clled focl chord. Focl chord perpendiculr to the xis is clled the ltus rectum of the prbol. 21. The eqution of prbol is simplest if the vertex is t the origin nd the xis of symmetry is long the x-xis or y-xis. The four possible such orienttions of prbol re shown below: 22. In terms of loci, the conic sections cn be defined s follows: Given line Z nd point F not on Z conic is the locus of point P such tht the distnce from P to F divided by the distnce from P to Z is constnt. i.e PF/PM = e, constnt clled eccentricity.
In cse of prbol eccentricity e =1. 23. Prbol is symmetric with respect to its xis. If the eqution hs y 2 term, thn the xis of symmetry is long the x-xis nd if the eqution hs n x 2 term, then the xis of symmetry is long the y-xis. 24. When the xis of symmetry is long the x-xis the prbol opens to the () Right if the coefficient of x is positive, (b) Left if the coefficient of x is negtive. 25. When the xis of symmetry is long the y-xis the prbol opens (c) Upwrds if the coefficient of y is positive. (d) Downwrds if the coefficient of y is negtive. 26. An ellipse is the set of ll points in plne, the sum of whose distnce from two fixed points in the plne is constnt. These two fixed points re clled the foci. For instnce, if F 1 nd F 2 re the foci nd P 1, P 2, P 3 re the points on the ellipse then P 1 F 1 +P 1 F 2 =P 2 F 1 +P 2 F 2 =P 3 F 1 +P 3 F 2 is constnt nd this constnt is more thn the distnce between the two foci. 27. An ellipse is the locus of point tht moves in such wy tht its distnce from fixed point (clled focus) bers constnt rtio, to its distnce from fixed line (clled directrix). The rtio e is clled the eccentricity of the ellipse. For n ellipse e <1.
28. The eccentricity is mesure of the fltness of the ellipse. The eccentricity of conic section is mesure of how fr it devites from being circulr 29. Terms ssocited with ellipse () The mid point of the line segment joining the foci is clled the centre of the ellipse. In the figure O is the centre of ellipse. For the simplest ellipse the centre is t origin. (b) The line segment through the foci of the ellipse is clled the mjor xis nd the line segment through the centre nd perpendiculr to the mjor xis is clled the minor xis. In the figure below AB nd In cse of simplest ellipse the two xes re long the coordinte xes. Two xes intersect t the centre of ellipse. (c) Mjor xes represent longer section of prbol nd the foci lies on mjor xes. (d) The end points of the mjor xis re clled the vertices of the ellipse. 30. If the distnce from ech vertex on the mjor xis to the centre be, then the length of the mjor xis is 2. Similrly, if the distnce of ech vertex on minor xis to the centre is b, the length of the minor xis is 2b. Finlly, the distnce from ech focus to the centre is c. So, distnce between foci is 2c.
31. Semi mjor xis, semi minor xis b nd distnce of focus from centre c re connected by the reltion 2 = b 2 + c 2 or c 2 = 2 b 2 32. In the eqution c 2 = 2 b 2, if is fixed nd c vry from 0 to, then resulting ellipses will vry in shpe. Cse (i) When c = 0, both foci merge together with the centre of the ellipse nd 2 = b 2, i.e., = b, nd so the ellipse becomes circle.thus circle is specil cse of n ellipse. Cse (ii) When c =, then b = 0. The ellipse reduces to the line segment F 1 F 2 joining the two foci. 33. The eccentricity of n ellipse is the rtio of the distnces form the centre of the ellipse to one of the foci nd to one of the vertices of the c ellipse. Eccentricity is denoted by e i.e., e. 34. The stndrd form of ellipses hving centre t the origin nd the mjor nd minor xis s coordinte xes. There re two possible orienttions:
35. Ellipse is symmetric with respect to both the coordinte xes nd cross the origin. Since if (x, y) is point on the ellipse, then (- x, y), (x, -y) nd (-x, y) re lso points on the ellipse. 36. Since the ellipse is symmetric cross the y-xis. It follows tht nother point F 2 (-c,0) my be considered s focus, corresponding to nother directrix. Thus every ellipse hs two foci nd two directrices. 37. The foci lwys lie on the mjor xis. The mjor xis cn be determined by finding the intercepts on the xes of symmetry. Tht is, mjor xis is long the x-xis if the coefficient of x 2 hs the lrger denomintor nd it is long the y-xis if the coefficient of y 2 hs the lrger denomintor. 38. Lines perpendiculr to the mjor xis A'A through the foci F1 nd F2 respectively re clled ltus rectum. Lines LL nd MM re ltus rectum. 39. The sum of focl distnces of ny point on n ellipse is constnt nd is equl to the mjor xis. 40. Conic ellipse cn be seen in the physicl world. The orbitl of plnets is ellipticl.
Aprt from this one cn see n ellipse t mny plces since every circle, viewed obliquely, ppers ellipticl. If the glss of wter is seen from top or if it is held stright it ppers to be circulr but if it is tilt it will be ellipticl. 41. A hyperbol is the set of ll points in plne, the difference of whose distnces from two fixed points in the plne is constnt. The two fixed points re clled the foci of the hyperbol. (Distnce to F 1 ) (distnce to F 2 ) = constnt 42. A hyperbol is the locus of point in the plne which moves in such wy tht its distnce from fixed point in the plne bers constnt rtio, e > 1, to its distnce from fixed line in the plne. The fixed point is clled focus, the fixed line is clled directrix nd the constnt rtio e is clled the eccentricity of the hyperbol.
43. Terms ssocited with hyperbol () The mid-point of the line segment joining the foci is clled the centre of the hyperbol. (b) The line through the foci is clled the trnsverse xis nd the line through the centre nd perpendiculr to the trnsverse xis is conjugte xis. (c)the points t which the hyperbol intersects the trnsverse xis re clled the vertices of the hyperbol. 44. The hyperbol is perfectly symmetricl bout the centre O. 45. Let the distnce of ech focus from the centre be c, nd the distnce of ech vertex from the centre be. Then, F 1 F 2 = 2c nd AB = 2 If the point P is tken t A or B then PF 2 -PF 1 =2 46. If the distnce between two foci is 2c, between two vertices is 2 i.e length of the trnsverse xis is 2, length of conjugte xis is 2b then,b,c re connected s c 2 = 2 + b 2 c 47. The rtio e is clled the eccentricity of the hyperbol. From the shpe of the hyperbol, we cn see tht the distnce of focus from
origin, c is lwys greter thn or equl to the distnce of the vertex from the centre, so c is lwys greter thn or equl to. Since c, the eccentricity is never less then one. 48. The simplest hyperbol is the one in which the two xes lie long the xes nd centre is t origin. Two possible orienttions of hyperbol re "north-south" opening hyperbol. Est-West Opening Hyperbol 49. A hyperbol in which = b clled n equilterl hyperbol. 50. Hyperbol is symmetric with respect to both the xes, since if (x, y) is point on the hyperbol. (-x, y), (x, -y) nd (-x, -y) re lso points on the hyperbol. 51. The foci re lwys on the trnsverse xis. Denomintor of positive term gives the trnsverse xis. 52. Ltus rectum of hyperbol is line segment perpendiculr to the trnsverse xis through ny of the foci nd whose end points lie on the hyperbol. Key Formule 1. The eqution of circle with centre (h, k) nd the rdius r is (x h) 2 + (y k) 2 = r 2. 2. If the centre of the circle is the origin O(0, 0), then the eqution of the circle reduces to x 2 + y 2 = r 2
3. y 2 = 4x y 2 = - 4x x 2 = 4y x 2 = - 4y Coordintes of vertex Coordintes of focus Eqution of the directrix Eqution of the xis Length of the Ltus Rectum (0,0) (0,0) (0,0) (0,0) (,0) (-,0) (0, ) (0, -) x = - x = y = - y = y = 0 y = 0 x = 0 x = 0 4 4 4 4 4.
Coordintes of the centre (0, 0) (0, 0) Coordintes of the vertices (, 0) nd (-, 0) (0, +b) nd (0, -b) Coordintes of foci (e, 0) nd (-e, 0) (0, be) nd (0, -be) Length of the mjor xis 2 2b Length of the minor xis 2b 2 Eqution of the mjor xis y = 0 x = 0 Eqution of the minor xis x = 0 y = 0 Equtions of the directrices Eccentricity 2 c b e 1 2 Length of the ltus rectum 2 2b c e 1 b b 2 2 b 2 2 5.
x 2 2 y 1 b 2 2 y b 2 2 x 1 2 2 Coordintes of the centre (0, 0) (0, 0) Coordintes of the vertices Coordintes of foci (, 0) nd (-, 0) (0, b) nd (0, -b) Length of the trnsverse xis Length of the conjugte xis Equtions of the directrices 2 2b 2b 2 Eccentricity 2 c b e 1 2 c e 1 b b 2 2 Length of the ltus rectum 2 2b 2 2 b Eqution of the trnsverse xis y = 0 x = 0 Eqution of the conjugte xis x = 0 y = 0