CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr lines through n point O in the plne of the pper. Point O is known s the origin. The line X'OX is lled the -is or is of ; the line 'O is known s the -is or is of nd the two lines tken together re lled the o-ordintes es or the es of o-ordintes. Qudrnt Nture of X nd Signs of o-ordinte XO I > 0 > 0 (+ +) OX' II < 0 > 0 (- +) X'O' III < 0 < 0 (- -) 'OX IV > 0 < 0 (+ -) Note - n point ling on -is or -is does not lie in n qudrnt. n point n e represented on the plne desried the o-ordinte es speifing its nd o-ordintes. The -o-ordinte of the point is lso known s the siss while the -oordinte is lso known s the ordinte. Distne Formul : The distne two point ( ) nd ( ) is given ( ) ( ) Note :. Distne is lws positive. Therefore we often write insted of.. The distne of point P ( ) from the origin. The distne etween two polr o-ordintes (r θ ) nd (r θ ) is given r r r r os(θ θ) pplition of Distne Formule : (i) For given three points C to deide whether the re olliner or verties of prtiulr tringle. fter finding C nd C we shll find tht the points re : Colliner - () If the sum of n two distnes is equl to the third i.e. + C = C. or + C = C or C + C = () If re of C is zero () If slope of = slope of C = slope of C. Verties of n equilterl tringle if = C = C Verties of n isoseles tringle if = C or C = C or C =. Verties of right ngled tringle if + C = C et. (ii) For given four points CD : = C = CD = D nd C = D CD is squre. = C = CD = D nd C D CD is rhomus. = CD C = D nd C = D CD is retngle. = CD C = D nd C D CD is prllelogrm. Note : The four given points re olliner if re of qudrilterl CD is zero. Digonls of squre rhomus retngle nd prllelogrm lws iset eh-other. Digonls of rhomus nd squre iset eh other t right ngle. Setion Formule :.The o-ordintes of point P( ) dividing the line segment joining the two points ( ) nd ( ) internll in the rtio m : m re given
m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the rtio m :m re given m m P m P m m m m m m m ( ) ( ) P ( ). The o-ordintes of the mid-point of the line segment joining the two points ( ) nd ( ) re given ( ) P ( ) ( ) Division es : If P ( ) nd Q ( ) then PQ is divided (i) - is in the rtio = (ii) - is in the rtio = Division Line : line + + = 0 divides PQ in the rtio = re of tringle : The re of tringle C whose verties re ( ) ( ) nd C( ) is denoted. Δ re of Polgon : The re of the polgon whose verties re ( ) ( )...( n n ) is -...... n n Some Importnt Points in Tringle : Centroid : If ( ) ( ) nd ( ) re the verties of tringle then the o-ordintes of its entroid re - Inentre : If ( ) ( ) nd C( ) re the verties of C s.t. C = C = nd = then the o-ordintes of its inentre re ( ) ( ) C ( ) Cirumentre : If ( ) ( ) nd C( ) re the verties of C then the oordintes of its irumentre re sin sin sin C sin sin sin C sin sin sin C sin sin sin C Orthoentre : Co-ordintes of orthoentre re tn tn tn C tn tn tn C tn tn tn C tn tn tn C Note : If the tringle is equilterl then entroid inentre orthoentre irumentre oinides. Orthoentre entroid nd irumentre re lws olliner nd entroid divides the line joining orthoentre nd irumentre in the rtio :. In n isoseles tringle entroid orthoentre inentre irumentre lies on the sme line. Inentre divides the ngles isetors in the rtio ( + ) : ( + ) : ( + ) :.
re of the tringle formed o-ordinte es nd the line + + = 0 is Stright Line : stright line is urve suh tht ever point on the line segment joining n two points on it lies on it. Different Forms of the Equtions of Stright Line : () Generl Form : The generl Form of the eqution of stright line is + + = 0 (First degree eqution in nd ). Where nd re rel onstnts nd re not simultneousl equl to zero. In this eqution slope of the line is given. The generl form is lso given = m + ; where m is the slope nd is the interept on -is. () Line Prllel to the X- The eqution of stright line to the -is nd t distne from it is given = Oviousl the eqution of the -is is = 0 () Line Prllel to - The eqution of stright line prllel to the -is nd t distne from is given = oviousl the eqution of -is is = 0 (d) Slope Interept Form : The eqution of stright line pssing through the point ( ) nd hving slope m is given ( - ) = m ( - ) (e) Two Points Form : The eqution of stright line pssing through two points ( ) nd ( ) is given ( - ) - Its slope (m) = (e) Interept Form : The eqution of stright line mking interepts nd on the es of nd respetivel is given -interept O X-interept X Slope (Grdient) of Line : m tn θ { + + = 0 = m + where m nd is onstnt } Here m is lled the slope or grdient of line nd is the interept on -is. The slope of line is lws mesured in ntilokwise. X O X Slope of line in terms of o-ordintes n two points on it :- If ( ) nd ( ) re o-ordintes of n two points on line then its slope is m X Differene of ordintes Differene of siss ngle etween two lines : m - m tn θ mm O C Condition of Prllellism of lines : If the slopes of two lines is m nd m nd if the re prllel then m m Length of Perpendiulr it or Distne of Point from Line : The length of perpendiulr from given point ( ) to line + + = 0 Note : The length of Perpendiulr from the t X D X
origin to the line + + = 0 is given Distne etween two Prllel Lines : If two lines re prllel the distne etween them will lws e the sme. When two stright lines re prllel whose equtions re + + = 0 nd + + = 0 then the distne etween them is given Chnges of es : If origin (0 0) is shifted to (h k) then the oordintes of the point ( ) referred to the old es nd (X ) referred to the new es n e relted with the reltion = X + h nd = + k O (hk) Point of Intersetion of Two Lines : Point of intersetion of two lines n e otined solving the equtions s simultneous equtions. If the given equtions of stright line re + + = 0 nd + + = 0 then (i) The ngle etween the lines θ is given tn θ. + (ii) If the lines re prllel then = 0 or X (iii) If the lines re perpendiulr then + = 0 (iv) Coinident : ngle etween lines os + sin = P nd os β + sin β = P is β Eerise --------------- LEVEL - ----------------. The point ( ) lies in the qudrnt : () First () Seond () Third (d) Fourth. The point ( ) lies in the qudrnt : () First () Seond () Third (d) Fourth. Find the distne etween the points ( 6) nd ( ) : () () () (d) 0. The distne etween the points (o) nd (0 ) () () () (d) +. The distne etween the points ( ) nd ( ) () 6 units () units () units (d) units 6. The o-ordintes of point situted on -is t distne of units from -is () (0 ) () ( 0) () ( ) (d) ( ). The o-ordintes of point situted on - is t distne of units from -is is : () (0 ) () ( 0) () ( ) (d) ( ) 8. The o-ordintes of point elow -is t distne of 6 units from -is ut ling on -is () (0 6) () ( 6 0) () (0 6) (d) (6 6) 9. The distne of the point (6 8) from the origin () units () units () units (d) 0 units 0. The point of intersetion of the lines + = nd + = () ( ) () ( ) () ( ) (d) ( ). The line + = meets -is t the point : () ( ) () (0 ) () ( 0) (d) ( 0). The line 9 = meets -is t the point : () 0 9 () 0 9 () 0 (d) 0. The slope of the line + + 8 = 0
() () () (d). The slope of the line joining P( ) nd Q( ) () () () (d). The eqution of line prllel to -is t distne of 6 units nd ove -is () = 6 () = 6 () = 6 (d) = 6 6. The eqution of line prllel to -is t distne of units to the left of -is is : () = () = () + = 0 (d) + = 0. The eqution of line prllel to -is nd t distne of units elow -is () = () = () = (d) = 8. The re of the tringle whose verties re P ( ) Q( 8) nd R ( ) (in squre units) () 66 () 6 () (d) 9. The points (0 0) (0 ) nd C( 0) re the verties of tringle whih () Isoseles () Right ngled () Equilterl (d) None of these 0. The o-ordintes of the entroid of PQR with verties P( 0) Q(9 ) nd R (8 ) 9 () ( 0) () 0 () (0 ) (d) ( 0). The eqution of line pssing through the points (0 ) nd ( ) () + + = 0 () + = 0 () + = 0 (d) = 0. The length of perpendiulr from the origin to the line + + = 0 () units () unit () units (d) units. The ngle whih the line joining the points nd mkes with is is : () 0 () () 60 (d) 90. The lines whose equtions re + = 0 nd 8 0 + 8 = 0 re : () prllel () oinident () perpendiulr (d) interseting --------------- LEVEL - ----------------. If the distne of the point P( ) from ( 0) is + then =? () () () 6 (d) 8. If the point ( ) is equidistnt from the points ( + ) nd ( + ) then =? () () () (d). If the sum of the squre of the distne of the point ( ) from the point ( 0) nd ( 0) is then : () + = + () + = () = + (d) + =. P ( ) nd Q( + ) re two points nd the o-ordintes of the middle point of PQ re ( ). The vlue of () 0 () () (d). If the points P( ) Q( ) nd R(6 ) re olliner the vlue of () / () / () 6 (d) 6. The eqution of line prllel to -is nd pssing through ( 6 ) () = () = 6 () = (d) = 6. The eqution of line prllel to -is nd pssing through ( ) () = () = () = (d) = 8. Two verties of tringle PQR re P( 0) nd Q( ) nd its entroid is ( 0). The o-ordintes of R re : () (8 ) () (8 ) () ( 8 ) (d) ( 8 ) 9. The o-ordintes of the point of intersetion of the medins of tringle with verties P(0 6) Q( ) nd R( ) re : () ( ) () ( ) () ( ) (d) ( ) 0. The rtio in whih the line segment joining ( ) nd ( ) is divided -is () : () : () : (d) 6 :. The rtio in whih the line segment joining P( ) nd Q ( ) is divided -is () : () : () : (d) : 0. The rtio in whih the point P divides the join of the point ( ) nd (
) () : () : () : (d) :. The eqution of line with slope nd pssing through the point ( ) () = + () = () = + (d) =. The vlue of so tht the lines + 8 = 0 nd + + = 0 re prllel () 0 () () (d). The vlue of P for whih the lines + 8 + 9 = 0 nd + p + 9 = 0 re perpendiulr () () 9 () (d) 9 6. The vlue of so tht line joining P( ) nd Q (0 ) nd the line joining ( ) nd (8 ) re perpendiulr to eh other () () () (d) 0. The ngle etween the lines represented the equtions 9 = 0 nd + = 0 () 0 () () 60 (d) 8. If P( ) Q ( ) nd R( 6) e n three points the ngle etween PQ nd PR () 0 () () 60 (d) 90 9. Given PQR with verties P ( ) Q ( ) nd R ( ). The eqution of medin PM () + 0 = 0 () 0 = 0 () + 0 = 0 (d) None of these 0. The o-ordintes of the point P whih di- vides the join of ( ) nd in the rtio : re : () ( ) () ( ) () (d) --------------- LEVEL - ----------------. The length of the portion of the stright line 8 + = 0 interepted etween the es () units () units () 6 units (d) units. The eqution of the line pssing through the point ( ) nd perpendiulr to the line + = 0 () + = 0 () + + k = 0 () = 0 (d) + = 0. The eqution of line pssing through the point ( ) nd prllel to the line + = 0 () = 0 () + = 0 () + = 0 (d) = 0. The sides PQ QR RS nd SP of qudrilterl hve the equtions + = = = + + = 0 respetivel then the ngle etween the digonls PR nd QS () 0 () () 60 (d) 90. The equtions of two equl sides of n isoseles tringle re + = 0 nd + = 0 nd its third side psses through the point ( 0). The eqution of the third side () = 0 ut not = 0 () neither + + = 0 nor = 0 () + + = 0 or = 0 (d) + + = 0 ut not = 0 6. If P nd P e perpendiulr from the origin upon the stright lines se θ + ose θ = nd os θ sin θ = os θ respetivel then the vlue of P P is : () () () (d). Find the eqution of the line pssing through the point ( ) nd utting off interepts on the es whose sum is 9? () + 6 = 0 ut not + 6 = 0 () neither + 6 = 0 nor + 6 = 0 () + 6 = 0 ut not + 6 = 0 (d) + 6 = 0 or + 6 = 0 LEVEL -. (). (d). (). (). (d) 6. (). () 8. () 9. (d) 0. (). (). (). (). (). (d) 6. (). () 8. () 9. () 0. (d). (). (). (). () LEVEL -. (). (). (d). (). () 6. (). () 8. () 9. () 0. (). (). (). (). (). () 6. (d). () 8. () 9. () 0. () LEVEL -. (d). (). (). (d). () 6. (). (d) 6
Hints nd Solutions : LEVEL -. () The point ( ) lies in the seond qudrnt.. (d) The point ( ) lies in the fourth qudrnt.. () Distne etween two points. () here ( ) = ( 6 ) nd ( ) ( ) Required distne = 6 68 6 unit 0 0. (d) = ( ) + ( ) = + = 6 + 9 = = units 6. () Clerl the point of -is hs ordinte 0 nd siss. So the point is ( 0). () Clerl the point on -is hs siss 0. So the point is (0 ) 8. () Clerl the point is (0 6) 9. (d) 0. () Required distne = 6 0 8 0 6 6 00 0 units + =... (i) + =...(ii) on solving (i) nd (ii) we get = nd = Required point of intersetion = ( ). () Eqution of -is is = 0 put = 0 in + = we get = Required point = ( 0). () Eqution of -is is = 0 put = 0 in 9 = we get = 9 Required point = 0 9. () + + 8 = 0 = 8 8. () Slope of Slope of PQ the line is 6. (d) Clerl; the eqution of the line is = 6 6. () Clerl the eqution of the line is =. () Clerl the eqution of the line is = 8. () 9. () 0. (d) 8 8 66 sq. units 0 0 0 C 0 0 0 nd C 0 0 + C = C C is right ngled tringle. (0 0) (0 ) C(0) The o-ordintes of the entroid of PQR re - 9 8 0 0. () The required eqution is 0 0 + = + + = o. () Length of perpendiulr = 0 0 units
. () The slope of the line is tn θ or θ 0. () Here 8 nd 0 8 So the given lines re o- inident. LEVEL -. () ( 0). () Let ( ) Q( + ) nd R( + ) re given points. PQ = PR. ( + ) + ( + ) + ( ) + (+ ) = + ( ) ( ) + + ( + ) ( + ) + + = + + = =.. (d) Let ( ) P( 0) nd Q( 0) Then P + Q = [( ) + ( 0) ] + [( + ) + ( 0) ] = + + + + + + = ( + + ) = + + = + =. (d) o-ordintes of middle point ( ) 8 =. () Sine PQ nd R olliner slope of PQ = slope of PR 6 8 = = 6 6. () The eqution of line prllel to -is is =. Sine it psses through ( 6 ) so = The required eqution is =. () The eqution of line prllel to -is is =. Sine it psses through ( ) so = The required eqution is = 8. () Let the o-ordintes of R e ( ). Then 0 nd 0 or + = nd + = 0 or = 8 nd = R = ( ) = (8 ) 9. () Sine point of intersetion of medin is entroid. o-ordintes of entroid 0 6 0. () Let the rtio e k : The ordinte of point ling on -is must e zero k 0 k k k Required rtio :. () Let the rtio e k : The iss of point ling on -is must e zero k 0 k 0 k k Required rtio is. () Let the rtio e k : : : k k k = k + k = k = Required rtio is :. () Let the eqution e = + Sine it psses through ( ) we hve = ( ) + = so its eqution is = +
. () Condition of prllelism lterntivel 8 8 0 m 0 0 m m m m tn θ m.m. So 8. () θ 0 Slope of PQ m 0 6 Slope of PR m for prllelism m = m. () Condition of perpendiulrism + = 0 + 8 p = 0 8p = p = 9 lterntivel- + 8 + 9 = 0 6. (d) 9 m 8 8 8 9 p 9 0 p p m p for perpendiulrism m.m = 8 p P 9 m = Slope of PQ 6 0 m = Slope of m m = + = = 0. () m 8 6. 9 0 0 9 9 m m tn θ m m 0 0 So 9. () θ Clerl M is the mid-point of QR. Co-ordintes of M re i.e. ( ) Now find the eqution of the line joining P( ) nd M( ) Required eqution is ( ) = ) 0 0 0. () Required point 0 ( LEVEL -. (d) Point of intersetion t -is = ( 0) 8 + = 0 8 + 0 = 0 = point ot intersetion = ( 0) Point of intersetion t -is = (0 ) 8 + = 0 0 + = 0 = 8 Point of intersetion = (0 8) Required length =
. () O (0 8) 0 0 8 6 units 89 (O) Given line - + = 0 its slope m Let m e the slope of required line. Then m m = or m = m = Let the required eqution e = m + Sine it psses through ( ) the required eqution is or = 0. () + = 0 its slope m Let the slope of line whih is prll;el to the given line is m m m Let the required eqution e Sine it psses through ( ) Required eqution is or 0. (d) + =... (i) + =...(ii) On solving (i) nd (ii) we get = = o-ordintes of P( ) S = + +=0 R P = + = Similrl Q( ) R( ) nd S( ) Q Now m = slope of PR = m = slope of QS = m m the required ngle is 90. () Third side psses through ( 0) so its eqution + 0 = m( )...(i) This side mkes equl ngle with the given two sides. let this ngle e θ. Now slope of line + = 0 is m m = nd slope of line + = 0 is m m = ngle etween (i) nd + = 0 = ngle etween (i) nd + = 0 m m tn θ m m m or / Hene possile equtions of third side re + 0 = ( ) nd 0 or + + = 0 nd = 0 6. () P = lenght of perpendiulr from (0 0) on se θ + ose θ = P se θ ose θ sinθ.osθ os θ sin θ or P = (sin θ. os θ ) P = sin θ 0
Similrl P os θ os θ sin θ os θ P P sin θ. os θ. (d) Let nd re the interepts on nd - es respetivel. + = 9 = 9...(i) nd the eqution of the line is...(ii) From (i) nd (ii)... iii 9 this line lso psses through the point () from (iii) 9 On solving we get = 6 or = If = 6 then = 9 6 = eqution of the line is 6 or + 6 = 0 If = then = 9 = 6 eqution of the line is 6 or + 6 = 0 Hene required eqution is + 6 = 0 or + 6 = 0 Note : Solve this tpe of question with the help of given options.