PARABOLA. moves such that PM. = e (constant > 0) (eccentricity) then locus of P is called a conic. or conic section.

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1 wwwskshieducioncom PARABOLA Le S be given fixed poin (focus) nd le l be given fixed line (Direcrix) Le SP nd PM be he disnce of vrible poin P o he focus nd direcrix respecively nd P SP moves such h PM or conic secion If e =, he conic is clled prbol If e <, he conic is clled n ellipse If e >, he conic is clled hyperbol = e (consn > 0) (eccenriciy) hen locus of P is clled conic 3 The generl equion of conic is S = x + hxy + by + gx + fy + c = 0 (Second degree equion in x nd y) i) If 0, h = b, hen S = 0 represens prbol ii) If iii) If 0, h <b, hen S = 0 represens n ellipse 0, h >b, hen S = 0 represens hyperbol iv) If 0, h >b, + b = 0, hen S = 0 represens recngulr hyperbol 4 sndrd forms of he prbol S No Conen I II III IV Equion y = 4x y = 4x x = 4y x = 4y Figure Y L L Y Z A S S A Z Y L L Y L Y S A Z Y L L Y Z A S Y L wwwskshieducioncom

2 wwwskshieducioncom Verex (A) (0, 0) (0, 0) (0, 0) (0, 0) Focus (S) (, 0) (, 0) (0, ) (0, ) 3 Poin of inersecionof xis nd direcrix (Z) (, 0) (, 0) (0, ) (0, ) 4 End poins of lus recum (L, L ) (, ) (, ) (, ) (, ) 5 Equion of xis y = 0 y = 0 x = 0 x = 0 6 Equion of Direcrix x = x = y = y = 7 Equion of ngen verex x = 0 x = 0 y = 0 y = 0 8 Equion of lus recum x = x = y = y = 9 Lengh of lusrecurm (LL ) Disnce from focus o direcrix (SZ) SA = AZ 5 i) If he xis of prbol is prllel o x-xis, equion of he prbol will be of he form (y ) = 4(x ) (or) (y ) = 4(x ) (or) x = y + by + c ii) If he xis of he prbol is prllel o y-xis, equion of he prbol will be of he form (x ) = 4(y ) (or) (x ) = 4(y ) (or) y = x + bx + c 6 In he equion of he prbol (y ) = 4(x ) i) Verex = (, ) ii) Focus = ( +, ) wwwskshieducioncom

3 wwwskshieducioncom iii) Ends of lusrecum = ( +, ) iv) Equion of xis is y = v) Equion of direcrix is x = vi)equion of lusrecum is x = + vii) Lengh of lusrecum = 4 7 In he equion of he prbol (x ) = 4(y ) i) Verex = (, ) ii) Focus = (, + ) iii) Ends of lusrecum = (, + ) iv)equion of xis is x = v) Equion of direcrix is y = vi)equion of lusrecum is y = + vii) Lengh of lusrecum = 4 8 The focl disnce of he poin P(x, y ) on he prbol i) y = 4x is SP = x + ii) x = 4y is SP = y + 9 A chord of he prbol perpendiculr o is xis is clled double or dine of he prbol 0 A chord of he prbol which is pssing hrough focus is clled focl chord The focl chord of he prbol which is perpendiculr o xis is clled lus recum Equion of Tngen o y = 4x (x, y ) is S = 0 wwwskshieducioncom

4 wwwskshieducioncom 3 The condiion h he line y = mx + c my be Tngen o he prbol y = 4x is c = m nd he poin of conc is m, m 4 The condiion for he line lx + my + n = 0 o be Tngen o prbol y = 4x is m = ln nd he poin of conc is n, l m l 5 The condiion h he line lx + my + n = 0 my be Tngen o he prbol x = 4y is l = mn nd he poin of conc is l n, m m 6 The equion of Tngen o y = 4x in slope from is y = mx + m x my + = 0 m (or) 7 If m nd m re he slopes of ngens from n exernl poin (x, y ) o he Prbol y = 4x hen hey re he roos of m x my + = 0 nd hence m + m = y x nd m m = x 8 If is he cue ngle beween Tngens drwn from (x, y ) o prbol S = y 4x = 0 hen Tn = S x 9 Locus of he poin of he inersecion of perpendiculr ngens drwn o he prbol is is direcrix 0 The ngle beween he ngens drwn from poin on he direcrix o be prbol is 90 o Tngens drwn he ends of focl chord of prbol re righ ngles, hey inersec on direcrix The ngle beween he ngens drwn he ends of lusrecum of he prbol y = 4x is 90 o nd he poin of inersecion of hese ngens is (, 0) wwwskshieducioncom

5 wwwskshieducioncom 3 The ngle beween he normls drwn he ends of lusrecum of he prbol y = 4x is 90 o nd he poin of inersecion of hese normls is (3, 0) 4 The Tngens nd normls he ends of lusrecum of prbol y = 4x forms squre whose side is nd re is 8 5 For he prbol y = 4(x + ), equion of ngen in slope from is y = m(x + ) + m 6 Equion of common ngen o wo prbols y = 4x nd x = 4by is /3 x + b /3 y + (b) /3 = 0 7 Equion of chord of conc of (x, y ) o y = 4x is S = 0 8 Pole of line lx + my + n = 0 w,r o prbol y = 4x is 9 Pole of line lx + my + n = 0 wr o prbol x = 4y is n m, l l l n, m m 30 The condiion for wo poins (x, y ) nd (x, y ) o be conjuge o prbol y = 4x is S = 0 3 The condiion h he lines l x + m y + n = 0 nd l x + m y + n = 0 o be conjuge wr o y = 4x is l n + l n = m m 3 The condiion h he lines l x + m y + n = 0 nd l x + m y + n = 0 o be conjuge wr o x = 4y is m n + m n = l l 33 If (x, y ) is midpoin of chord of y = 4x hen equion of chord is S = S 34 For ny curve y = f(x), he slope of chord hving (x, y ) s middle poin is dy dx x,y 35 Equion of pir of ngens drwn from n exernl poin (x, y ) o prbol y = 4x is S = SS wwwskshieducioncom

6 wwwskshieducioncom Prmeric Form: 36 For ll vlues of, he poin (, ) lies on prbol y = 4x nd i is denoed by Equions x =, y = re clled prmeric equion of y = 4x 37 The focl disnce of poin P(, ) on he prbol y = 4x is + 38 If (, ) is one end of he double ordine of y = 4x hen is lengh = 4 39 Equion of chord joining nd on prbol y = 4x is y( + ) = x + 40 If, re he ends of he focl chord of y = 4x, hen = 4 If (, ) is one end of focl chord of he prbol y = 4x hen is oher end is, 4 Slope of he ngen on y = 4x is 43 Equion of ngen o y = 4x is y = x + 44 The poin of inersecion of ngens nd on y = 4x is [, ( + )] 45 Slope of he norml on y = 4x is 46 Equion of norml on y = 4x is y + x = + 3 Since his is cubic equion in, i hs 3 roos in which les one of hem is rel Therefore from given poin, we cn drw mos hree normls o prbol 47 If,, 3 re he fee of he normls drwn from poin (x, y ) o he prbol y = 4x hen hey re he roos of 3 + ( x ) y = 0 nd hence = 0, =, 3 = y - x 48 The ngen end of focl chord of prbol is prllel o norml he oher end 49 Equion of norml o y = 4x in slope form is y = mx m m 3 where m is slope of norml wwwskshieducioncom

7 wwwskshieducioncom 50 The condiion h he line lx + my + n = 0 o be norml o prbol y = 4x is l 3 + lm + m n = 0 5 If he norml on y = 4x mees i gin hen = 5 If he normls nd d on he prbol y = 4x mee gin on prbol 3 hen = nd = 0 53 If he norml chord on y = 4x subends righ ngle he verex hen = 54 If he norml chord on y = 4x subends righ ngle he focus hen = 4 Lengh of Chord: 55 If P(x, y ), Q(x, y ) re he ends of chord of curve hen is lengh PQ = x x m where m is slope of PQ 56 If nd re ends of chord of y = 4x hen is lengh = 4 57 The lengh of focl chord drwn poin on he prbol y = 4x is 58 If focl chord of prbol y = 4x mkes n ngle wih is xis hen is lengh = 4 cosec 4 59 Lengh of norml chord on y = 4x is 3/ 60 Lengh of chord of conc of (x, y ) wr o y = 4x is S y 4 6 Lengh of he chord of y = 4x hving (x, y ) s is mid poinis S y 4 wwwskshieducioncom

8 wwwskshieducioncom Are of he Tringle: 6 Are of ringle formed by wo ngens from (x, y ) nd is chord of conc is 3/ S 63 Are of ringle inscribed in prbol y = 4x is 8 y y y y3 y3 y y y nd y 3 re ordines of ngulr poins where 64 Are of he ringle formed by he ngens hree poins whose he ordines y, y, y 3 on y = 4x is y y y y3 y3 y 6 Exr Informion: 65 If PSQ is focl chord of conic hen SP SQ l when l is lengh of semi lusrecum of conic (Or) The semi lus recum is hrmonic men beween segmens of focl chord of he prbol 66 If SP nd SQ re he disnces of wo poin P nd Q on prbol y = 4x from focus S, T is pin on inersecion of ngen P nd Q hen S = SP SQ 67 Les lengh of focl chord of y = 4x is 4 68 The lengh of he norml chord of he prbol is les when i subends righ ngle he verex 69 The orhocenre of he ringle formed by hree ngens of prbol lies on direcion 70 The circle pssing hrough hree fee of he normls drwn from poin o he prbol wwwskshieducioncom

9 wwwskshieducioncom 7 The circum circle of he ringle formed by ny hree ngens o prbol psses hrough he focus of he prbol 7 P is poin on he prbol whose focus is S nd PN is he perpendiculr drwn from P o he direcrix, hen he ngen P is he inernl bisecor P nd norml is he exernl bisecor of P wwwskshieducioncom

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