THREE DIMENSIONAL GEOMETRY

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1 MD THREE DIMENSIONAL GEOMETRY CA CB C Coordintes of point in spe There re infinite numer of points in spe We wnt to identif eh nd ever point of spe with the help of three mutull perpendiulr oordintes es OX OY nd OZ Vetor representtion of point in spe If oordinte of point P in spe is ( ) then the position vetor of the point P with respet to the origin is î ĵ kˆ Distne formul C Distne etween n two points ( ) nd ( ) is given s ( ) ( ) ( Distne of point P from oordintes es ) Let PA PB nd PC re distnes of the point P( ) from the oordinte es OX OY nd OZ respetivel then PA PB PC C Setion Formul If point P divides the distne etween the points A( ) nd B ( ) in the rtio of m : n then oordintes of P re given s Note : Mid point is m n m n m n m n m n m n C5 C6 Centroid of tringle ABC G Inentre of tringle ABC C C8 Centroid of tetrhedron A ( ) B ( ) C ( ) nd D ( ) re the verties of tetrhedron then oordintes of the entroid is i i i Diretion osines nd diretion rtios (i) Diretion osines : Let e the ngles whih direted line mkes with the positive diretions of the es of nd respetivel then os os os re lled the diretion osines of the line The diretion osines re usull denoted (l m n) (ii) If l m n e the diretion osines of line then l + m + n = (iii) Diretion rtios : Let e proportionl to the diretion osines l m n then re lled the diretion rtios

2 (iv) If re the diretion rtios of n line L then L î MD ĵ kˆ will e vetor prllel to the line If l m n re diretion osines of line L then l î mĵ nkˆ is unit vetor prllel to the line L If l m n e the diretion osines nd e the diretion rtios of vetor then l m n (v) (vi) (vii) or l m If OP = r when O is the origin nd the diretion osines of OP re l m n then the oordintes of P re (lr mr nr) If diretion osines of the line AB re l m n AB = r nd the oordintes of A is ( ) then the oordintes of B is given s ( + rl + rm + rn) If the oordintes P nd Q re ( ) nd ( ) then the diretion rtios of line PQ re = = & = nd the diretion osines of the line PQ re l m nd n PQ PQ PQ Diretion osines of es : Sine the positive -is mkes ngles with es of nd respetivel Therefore Diretion osines of -is re ( 0 0) Diretion osines of -is re (0 0) Diretion osines of -is re (0 0 ) C9 Angle etween two line segments : If two lines hve diretion rtios nd respetivel then we n onsider two vetors prllel to the lines s i + j + k nd i + j + k nd ngle etween them n e given s os (i) The line will e perpendiulr if + + = 0 n (ii) The lines will e prllel if (iii) Two prllel lines hve sme diretion osines ie l = l m = m n = n Prtie Prolems : Find the diretion osines of the line whih re onneted the reltions l 5m + n = 0 nd l + 5m n = 0 Prove tht the stright lines whose diretion osines re given the reltions l + m + n = 0 nd f g h fmn + gnl + hlm = 0 re perpendiulr to eh other if 0 nd prllel if f + g + h gh hf fg = 0 Show tht the stright lines whose diretion osines re given the equtions l + m + n = 0 nd ul + vm + wn = 0 re mutull perpendiulr if (v + w) + (u + w) + (u + v) = 0 nd prllel if 0 u v w

3 Show tht the ngle etween n two digonls of ue is os MD 5 A line mkes ngles with the four digonls of ue Prove tht os + os + os + os = 6 If vrile line in two djent positions hs diretion osines (l m n) nd (l + l m + m n + n) show tht the smll ngle etween the two positions is given () = (l) + (m) + (n) If l m n nd l m n e the diretion osines of two mutull perpendiulr lines show tht the diretion osines of the line perpendiulr to oth of them re (m n m n ) (n l n l ) (l m l m ) [Answers : () nd 6 6 ] 6 C0 C Projetion of line segment on line (i) (ii) If the oordintes P nd Q re ( ) nd ( ) then the projetion of the line segment PQ on line hving diretion osines l m n is l( ) + m ( ) + n( ) Vetor form : projetion of vetor on nother vetor is In the ove se we n onsider PQ s ( ) î + ( ) ĵ + ( ) kˆ in ple of nd l î mĵ nkˆ in ple of (iii) l r m r &n r re the projetion of r in OX OY & OZ es (iv) r r ( lî mĵ nkˆ ) A PLANE If line joining n two points on surfe lies ompletel on it then the surfe is plne OR If line joining n two points on surfe is perpendiulr to some fied stright line Then this surfe is lled plne The fied line is lled the norml to the plne Eqution of plne (i) (ii) Norml form of the eqution of plne is l + m + n = p where l m n re the diretion osines of the norml to the plne nd p is the distne of the plne from the origin Generl form : d = 0 is the eqution of plne where re the diretion rtios of the norml to the plne (iii) The eqution of plne pssing through the point ( ) is given ( ) + ( ) + ( ) = 0 where re the diretion rtios of the norml to the plne (iv) Plne through three points : The eqution of the plne through three non-olliner points ( ) ( ) ( ) is 0 (v) Interept Form : The eqution of plne utting interept on the es is

4 (vi) Note : () () MD Vetor Form : The eqution of plne pssing through point hving position vetor nd norml n is (r )n 0 or rn n Vetor eqution of plne norml to unitvetor nˆ nd t distne d from the origin is rn d Coordinte plnes (i) Eqution of -plne is = 0 (ii) Eqution of -plne is = 0 (iii) Eqution of -plne is = 0 () Plnes prllel to the es : If = 0 the plne is prllel to -is ie eqution of the plne prllel to the -is is + + d = 0 Similrl eqution of plnes prllel to -is nd prllel to -is re + + d = 0 nd + + d = 0 respetivel Plne through origin : Eqution of plne pssing through origin is + + = 0 (e) Trnsformtion of the eqution of plne to the norml form : To redue n eqution + + d = 0 to the norml form first write the onstnt term on the right hnd side nd mke it positive then divide eh term where re oeffiients of nd respetivel eg (f) (g) (h) (i) (j) where (+) sign is to e tken if d > 0 nd ( ) sign is to e tken if d < 0 An plne prllel to the given plne d = 0 is = 0 Distne etween two prllel plnes d = 0 nd d = 0 is given s d d Eqution of plne pssing through given point & prllel to the given vetors : The d eqution of plne pssing through point hving position vetor nd prllel to & is r µ (prmetri form) where & µ re slrs or r( ) ( ) (non prmetri form) A plne d = 0 divides the line segment joining ( ) nd ( ) in the rtio d d The -plne divides the line segment joining the points ( ) nd ( ) in the rtio Similrl -plne in Coplnrit of four points nd -plne in The points A( ) B( ) C( ) nd D( ) re oplnr then 0

5 MD 5 Ver similr in vetor method the point A(r )B(r )C(r ) nd D(r ) re oplnr if r r r r r r ] 0 [ Prtie Prolems : A vrile plne moves in suh w tht the sum of the reiprols of its interepts on the oordintes es is onstnt Show tht the plne psses through fied point Find the vetor eqution of plne whih is t distne of 6 units from the origin nd whih hs ĵ s the unit vetor norml to it Find the Crtesin eqution of plne whose vetor eqution is r (î 5ĵ kˆ ) Find the vetor eqution of plne whose Crtesin eqution is 5 + = 5 Redue the eqution of the plne + 9 = 0 to the norml form nd hene find the length of the perpendiulr from the origin to the plne Also find the diretion osines of the norml to the plne 6 Find the vetor eqution of the plne through ( ) whih is prllel to the plne r (î ĵ 5kˆ ) 0 Find the eqution of the plne pssing through the line of intersetion of the plnes + = nd = 0 nd the point ( ) 8 Find the eqution of the plne whih ontins the line of intersetion of the plnes + + = 0 nd = 0 nd whih is perpendiulr to the plne = 0 9 Find the eqution of plne through the intersetion of the plnes r (î ĵ kˆ ) 5 nd r (î ĵ kˆ ) nd pssing through the point ( ) 0 Find the vetor eqution of the plne through the line of intersetion of the plnes r (î ĵ kˆ ) nd r (î ĵ) 0 nd perpendiulr to the plne r (î ĵ kˆ ) 8 0 Find the Crtesin s well s the vetor eqution of the plne through the intersetion of the plnes r (î 6ĵ) 0 nd r (î ĵ kˆ ) 0 whih is t unit distne from the origin Find the distne etween the prllel plnes r(î ĵ 6kˆ ) 5 nd r(6î 9ĵ 8kˆ ) 0 0 [Answers : () k k k () r ĵ 6 () + 5 = () r (5î ĵ kˆ ) (5) (6) r (î ĵ 5kˆ ) 0 () = 0 (8) = 0 (9) r (î ĵ) 8 (0) r ( 5î ĵ kˆ ) 5 () = 0 nd + = 0 () units ] C Sides of plne : C A plne divides thre three dimensionl spe in two equl prts Two points A ( ) nd B ( ) re on the sme side of the plne d = 0 if d nd d re oth positive or oth negtive nd re opposite side of plne if oth of these vlues re in opposite sign A plne & point (i) Distne of the point ( ) from the plne d = 0 is given d

6 MD 6 r n (ii) The length of the perpendiulr from point hving positive vetor to plne d is given n d p n (iii) (iv) () The oordintes of the foot of perpendiulr from the point ( ) to the plne d = 0 re given To find imge of point wrt plne ( d) Let P ( ) is given point nd d = 0 is given plne Let ( ) is the imge point then () 0 Solve out to get ( ) The oordintes of the imge of point ( ) wrt the plne d = 0 re given ( Prtie Prolems : Find the imge of the point P( ) in the plne + + = 0 A vrile plne whih remins t onstnt distne p from the origin uts the oordintes es t A B C Show tht the lous of the entroid of ABC is + + = p A vrile plne is t onstnt distne p from the origin nd meets the oordintes es in A B C Show tht the lous of the entorid of the tetrhedron OABC is + + = 6p [Answers : () Q( 5 )] d) C Angle etween two plnes : (i) Consider two plnes d = 0 nd d 0 Angle etween these plnes is the ngle etween their normls Sine diretion rtios of their normls re ( ) nd ( ) respetivel hene the ngle etween them is given os Plnes re perpendiulr if 0 nd plnes re prllel if nn (ii) The ngle etween the plnes rn d nd rn d is given os n n Plnes re perpendiulr if nn 0 & plnes re prllel if n n Prtie Prolems : Find the ngle etween the plnes + = nd + = Find the vlue of for whih the plnes + = nd + + = 8 re perpendiulr to eh other 5 [Answers : () os () = ] 58

7 MD C5 Angle isetors (i) The equtions of the plnes iseting the ngle etween two given plnes d = 0 nd d = 0 re d d gives the isetor of the ngle whih ontins the origin (ii) Eqution of isetor of the ngle ontining origin First mkes oth the onstnt terms positive Then the positive sign in d d gives the isetor of the ngle whih ontins the origin (iii) Bisetor of ute/otuse ngle : First mke oth the onstnt terms positive Then + + > 0 origin lies on otuse ngle + + < 0 origin lies in ute ngle C6 Fmil of plnes (i) An plne pssing through the line of intersetion of non-prllel plnes or eqution of the plne through the given line in severl form d = 0 & d = 0 is d + ( d ) = 0 (ii) The eqution of plne pssing through the intersetion of the plnes d d ) n r (n is d & rn d n r where is ritrr slr C Are of tringle : Let A( ) B( ) C( ) e the verties of tringle then re of the tringle is ) ( where nd Vetor Method From two vetors AC nd AB Then re is given k j i AC AB C8 Volume of tetrhedron : Volume of tetrhedron with verties A( ) B( ) C( ) nd D( ) is given 6 V A LINE C9 Eqution of line (i) A stright line in spe is hrterised the intersetion of two plnes whih re not prllel nd therefore the eqution of stright line is solution of the sstem onstituted the equtions of the two plnes d = 0 nd d = 0 This form is lso known s non-smmetril form

8 MD 8 (ii) The eqution of line pssing through the point ( ) nd hving diretion rtios (iii) (iv) is r This form is lled smmetri form A generl point on the line is given ( + r + r + r) Vetor eqution : Vetor eqution of stright line pssing through fied point with position vetor nd prllel to given vetor is r where is slr The eqution of the line pssing through the points ( ) nd ( ) is (v) Vetor eqution of stright line pssing through two points with position vetors & is r ( ) (vi) Note : Redution of rtesion form of eqution of line to vetor form & vie vers Stright lines prllel to o-ordintes es : ( î ĵ kˆ ) (î ĵ kˆ ) r Stright lines Equtions Stright lines Eqution (i) Through origin = m = n (v) Prllel to -is = p = q (ii) -is = 0 = 0 (vi) Prllel to -is = h = q (iii) -is = 0 = 0 (viii) Prllel to -is = h = p (iv) -is = 0 = 0 Prtie Prolems : Find the vetor eqution of line pssing through point with the position vetor ( î ĵ kˆ ) nd prllel to the line joining the points ( î ĵ kˆ ) nd ( î ĵ kˆ ) Also find the Crtesin equivlents of the eqution The Crtesin equtions of line re 6 Find the vetor eqution of the line 5 The Crtesin equtions of line re 6 = + = Find () the diretion rtios of the line nd () the Crtesim nd vetor equtions of the line prllel to this line nd pssing through the point ( ) Find the oordintes of the point where the line through A( ) nd B(5 6) rosses the -plne 5 5 Find the oordintes of the point where the line meets the plne + = 6 Find the equtions of the line pssing through the point ( ) nd perpendiulr to eh of the line nd 5 Show tht the lines r (î ĵ kˆ ) (î ĵ) nd r (î kˆ ) µ(î kˆ ) interset eh other Find their point of intersetion 8 Prove tht the points A( ) B(5 0 5) nd C( ) re olliner 9 Find the vlue of for whih the points A( ) B( ) nd C(5 5 ) re olliner 0 Show tht the points whose position vetors re ( î ĵ 5kˆ )(î ĵ kˆ ) nd (î kˆ ) re olliner

9 MD 9 Using the vetor method find the vlues of nd µ for whih the points A( µ) B( 0 ) nd C( 5) re olliner [Answers : () () r (î ĵ 6kˆ ) (î 5ĵ kˆ ) () r (î ĵ kˆ ) (î ĵ kˆ ) () ( 0 ) () = nd µ = ] () (5) P( ) (6) C0 Redution of non-smmetril form to smmetril form : Let eqution of the line is non-smmetril form e d = d = 0 To find the eqution of the line in smmetril form we must know (i) its diretion rtios (ii) oordintes of n point on it (i) Diretion rtios : Let l m n e the diretion rtios of the line Sine the line lies in oth the plnes it must e perpendiulr to normls of oth plnes So l + m + n = 0 l + m + n = 0 From these equtions proportionl vlues of l m n n e found rossmultiplition s l Alterntive method The vetor m i j k n = i( ) + j( ) + k( ) will e prllel to the line of intersetion of the two given plne hene l : m : n = ( ) : ( ) : ( ) (ii) Point of the line Note tht s l m n nnot e ero simultneousl so t lest one must e non-ero Let 0 then the line nnot e prllel to plne so it interset it Let it interset -plne in ( 0) Then + + d = 0 nd + + d = 0 Solving these we get point on the line Then its eqution eomes 0 d d or d d 0 Note : If l 0 tke point on -plne s (0 ) nd if m 0 tke point on -plne s ( 0 ) Alterntive method C If Put = 0 in oth the equtions nd solve the equtions + + d = d = 0 otherwise put = 0 nd solve the equtions + + d = 0 nd + + d = 0 Foot length nd eqution of perpendiulr from point to line L (i) Crtesin form : Let eqution of the line e r (s) l m n nd A( ) e the point An point on line (i) is P (lr + mr + nr + ) If it is the foot of the perpendiulr from A on the line then AP is perpendiulr to the line So l(lr + ) + m(mr + ) + n (nr + ) = 0 ie r = n( )l + ( )m + ( )n sine l + m + n = Putting this vlue of r in (ii) we get the foot of perpendiulr from point A on the given line Sine foot of perpendiulr P is known then the length of perpendiulr is given (i) (ii)

10 AP MD 0 ( l r ) (mr ) (nr ) the eqution of perpendiulr is given lr mr nr C (ii) Vetor Form : Eqution of line pssing through point hving position vetor nd perpendiulr so the lines r nd r is prllel to So the vetor eqution of suh line is ( ) Position vetor of the imge of point in r ( ) stright line r is given Position vetor of the foot of ( ) the perpendiulr on line is f The eqution of the perpendiulr is ( ) r µ ( ) To find imge of point wrt line Let L is given line Let ( ) is the imge of the point P ( ) with respet to the line L Then (i) ( ) ( ) ( ) 0 (ii) from (ii) get the vlue of in terms of s now put the vlue of Prtie Prolems : in (i) get nd resutitute the vlue of to get ( ) Find the imge of the point ( 6 ) in the line [Answers : () M( 0 )] C Angle Between A Plne And A Line : (i) If is the ngle etween line l m n nd the plne d = 0 then sin ( l m n ) l m n

11 (ii) Vetor form : If is the ngle etween line r ( ) nd rn d then MD n sin n (iii) l m n Condition for perpendiulrit n 0 (iv) Condition for prllel l + m + n = 0 n 0 Prtie Prolems : Find the ngle etween the line r (î ĵ kˆ ) (î ĵ kˆ ) nd the plne r(6î ĵ kˆ ) 5 Find the vlue of m for whih the line r (î ĵ kˆ ) (î ĵ kˆ ) is prllel to the plne r(î ĵ mkˆ ) Show tht the line r (î ĵ kˆ ) (î ĵ kˆ ) is prllel to the plne r(î 5ĵ kˆ ) 5 Also find the distne etween them Find the ngle etween the line nd the plne = 0 [Answers : () 8 sin 0 () () units () sin ] 5 C Condition For A Line To Lie In A Plne (i) Crtesin form : Line l m n would lie in plne d = 0 if d = 0 & l + m + n = 0 (ii) Vetor form : Line r would lie in the plne rn d if n 0 & n d C5 Coplnr Lines : (i) If the given lines re l m n nd then ondition l m n for intersetion/oplnrit is l m n 0 & plne ontining the ove two l m n lines is l m n 0 l m n (ii) Condition of oplnrit if oth the lines re in generl form let the lines e d = 0 = d & = 0 = d d re oplnr if 0

12 Prtie Prolems : MD Find the eqution of the plne pssing through the point (0 ) nd ontining the line Find the eqution of the plne pssing through the line of intersetion of the plnes + = nd + + = 8 nd prllel to the line with diretion rtios Find lso the perpendiulr distne of ( ) from this plne Find the eqution of the plne pssing through the line of intersetion of the plnes + = = 0 nd prllel to the line 5 5 Find the eqution of the plne pssing through ( ) nd ( ) nd prllel to the -is 5 Show tht the lines r (î ĵ kˆ ) (î ĵ) nd r (î kˆ ) µ(î kˆ ) re oplnr Also find the eqution of the plne ontining oth these lines 6 6 Show tht the lines nd re oplnr Also find the eqution of the plne ontining these lines d d Show tht the lines nd re oplnr 8 Find the eqution of the plne whih ontins the two prllel lines nd [Answers : () + + = 0 () units () = 0 () + 5 = 0 9 (5) r(î 9ĵ kˆ ) 0 (6) + + = 0 (8) = 0] C6 Skew Lines : (i) The stright lines whih re not prllel nd non-oplnr ie non-interseting re lled skew lines If l m n 0 then lines re skew l m n (ii) Shortest distne : Suppose the eqution of the lines re l m n nd l m n then shortest distne is given ( )(mn mn) ( )(nl nl) ( )( lm lm) (mn mn) = l m n (mn mn) l m n

13 (iii) Vetor Form : For lines & to e skew )( ) 0 or ( )] 0 ( [ (iv) Shortest distne etween the two prllel lines & ( ) r µ is d r MD Prtie Prolems : Find the shortest distne etween the lines whose vetor equtions re r ( t)î (t )ĵ ( t)kˆ nd r (s )î (s )ĵ (s )kˆ Find the length nd the equtions of the line of shortest distne etween the lines 8 nd 6 Show tht the lines nd 5 interset eh other Find their point of intersetion Find the distne of the point ( ) from the plne = 0 mesured prllel to the line 6 5 Find the distne of the point ( ) from the line the plne + + = 0 5 mesured prllel to [Answers : () units () () ( ) (5) / units] C Sphere Generl eqution of sphere is given u + v + w + d = 0 The entre of the sphere is ( u v w) nd rdius is u v w d

14 The re of qudrilterl ABCD where A (0 ) B ( ) C ( 5 0) nd D ( 6 ) is equl to () 9 sq units () 8 sq units () sq units 8 sq units The ngle etween lines whose diretion osines re given l + m + n = 0 l + m n = 0 is () () 6 () None of these If the diretion osines of vrile line in two djent positions e l m n nd l + dl m + dm n + dn nd the smll ngle etween the two positions is d then () d = dl + dm + dn () = (dl) + (dm) + (dn) () d = dl + dm + dn None of these The stright lines k 5 will interset provided k () k = { } () k = {0 } () k = { } k = {0 } nd 5 The plne + + = d meets the oordinte es t the points A B nd C respetivel Are of tringle ABC is equl to () () () d d d None of these 6 The distne of the point ( ) from the plne + 5 = 0 mesured prllel to the line is equl to 6 () unit () units () units None of these MD SINGLE CORRECT CHOICE TYPE Two sstems of retngulr es hve the sme origin If plne uts them t distnes nd from the origin then () () () + + = = + 8 Let f() e polnomil in stisfing the ondition f ()f f() + f nd f() = 5 The diretion osines of the r joining origin nd point (f(0) f() f()) is () () () If origin is the entroid of ABC with verties A( ) B( 5) nd C( ) the vlues of nd is () = = 8 = () = = 8 = () = = 8 = = = 8 = 0 Let f e one-one funtion with domin ( 0) nd rnge { } suh tht etl one of the following sttements is true f( ) = f() f(0) nd the remining two re flse The distne etween points ( 0) nd (f( ) f() f(0)) is () 8 () () 6 5 A( 0 ) B(0 ) C( ) re three points nd D is the foot of perpendiulr from A to BC The o-ordintes of D is () ( 5 5) () ( 5 5) () ( 5 5) ( 5 5)

15 The imge of the point ( ) with respet to the plne + + = 0 is () () () A vrile plne psses through fied point ( ) nd meets the oordintes is in A B C The lous of mid point of the plne ommon through A B C nd prllel to the oordinte plnes is () + + = () + + = () + + = none of these The imge of the point P( ) the plne l + m + n = 0 is Q ( ) then MD 5 ANSWERS (SINGLE CORRECT CHOICE TYPE) 9 d 0 d d d () + + = l + m + n () + + = ( ) ( ) ( ) () 0 l( ) m( ) n( ) 0 5 A squre ABCD of digonl units is folded long the digonl AC so tht plne DAC BAC re t right ngle the shortest distne etween DC nd AB is () / () / () none of these

16 COMREHENSION TYPE COMPREHENSION- A r of light emnipting from the point soure P( ) nd trvelling prllel to the line is inident on the plne + = 0 t the point Q After refleting from the plne the r trvels long the line QR It is lso known tht the inident r refleted r nd the norml to the plne t the point of inidene re in the sme plne The point Q is () ( 5 6) () ( 6 ) () ( 6 ) ( 5 6) The eqution of the line QR re MD 6 EXCERCISE BASED ON NEW PATTERN COMPREHENSION- Consider the two lines L nd L whose equtions re nd respetivel Consider the plne + + = Choose the orret sttement () () () oth lines re prllel to the plne onl L is prllel to the plne onl L is prllel to the plne none of these 8 The -oordinte of point of intersetion of one of ove lines with the plne is () () 0 () () () () 9 The line whih interset the plne mkes n ngle of () sin 5 () os 5 6 The eqution of the plne PQR is () = 0 () 5 + = 0 () 5 = 6 + = 6 COMPREHENSION- ABC is tringle where A = ( 5) B = ( ) nd C = ( 5 µ) The medin through A is equll inlined to the es The vlue of is () () 5 () 0 5 The vlue of µ is () () 5 () 0 6 The re of tringle ABC is () () 8 () () tn none 5 COMPREHENSION- P( 5) is point nd + + = 0 is plne 0 The length of the perpendiulr from P to the plne is () () () The -oordinte of foot of the perpendiulr from P to the plne is () () 8 6 () none

17 The oordinte of imge of P with respet to the plne is () () 5 () none COMPREHENSION-5 Consider the points A( 5) nd B(0 6 ) nd the plne + = Choose the orret sttement from the following () oth points re on the sme side of the plne () oth points re on the opposite sides of the plne () oth points re in the plne none of these The rtio in whih the line segment AB is divided the plne is () : 5 internll () : 5 eternll () 0 : internll 0 : eternll 5 If P( + ) is point on the sme side of the plne s the point A then the set of vlues of is () () () COMPHENSION-6 Consider the lines L nd L nd 6 The shortest distne etween them is () () 9 9 () MD The equtions of the line of shortest distne () () () 5 5 none The -oordinte of the point of intersetion of the ove line with the line L is () () 58 5 MATRIX-MATCH TYPE Mthing- Column - A () Column - B (A) A line mkes ngles (P) / with the positive diretions of the es of referene The vlue of os + os + os is (Q) (B) The diretion rtios of two perpendiulr lines re 5 nd + + Then is (C) The ngle etween the (R) 6 lines joining the points ( 0) ( + ) nd (0 0) ( ) is os If is n integer 8 then its vlue is (D) The volume of the (S) 0 tetrhedron whose verties re (0 ) ( 6) ( ) nd ( 0 ) is (in unit ) Mthing- Column - A Column - B (A) If A = ( ) (P) B = ( ) nd AOB = / where O is the origin then is

18 (B) If the points A( ) (Q) B( 6 ) nd C( ) re olliner then is (C) If A = ( ) (R) 0 B = ( ) C = (0 ) D = ( ) nd AB AC nd AD re oplnr then is (D) The projetion of line (S) segment on the es of referene re nd respetivel The length of the line segment is Mthing- Column - A Column - B (A) The lines (P) 6 nd re perpendiulr to eh other Then is equl to (B) The shortest distne (Q) infinite etween the line 0 nd the -is is (C) The projetion of the (R) line segment joining the point (6 ) nd the origin on the line 0 is (D) The numer of rel vlues (S) /5 of k for whih the lines k nd (k ) re interseting is Mthing- Column - A Column - B (A) The distne of the (P) /5 point ( 0 ) from the plne 5 = 0 is (B) The distne etween (Q) / the plnes 5 + = 5 nd = 0 is MD 8 (C) The shortest distne (R) 0/ etween the lines = 0 = + nd + = 0 = + is (D) A vrile plne t (S) 9 distne of unit from the origin uts the oordinte es t A B nd C If the entroid D( ) stisfies the reltion + + = k then the vlue of k is MULTIPLE CORRECT CHOICE TYPE A point Q t distne from the point P( ) ling on the line joining the points A(0 ) nd P hs the oordintes () ( ) () ( 5) () (0 ) ( 5 ) The diretion osines of the line pssing through the origin nd utting the line t () () () os 6 re The eqution of plne is = 5 nd A( ) B( ) C( ) nd D( ) re four points Whih of the following line segments re interseting the plne? () AD () AB () AC BC Whih of the following is/re orret out tetrhedron () () () Centroid of tetrhedron lies on lines joining n verte to the entroid of oppsite fe Centroid of tetrhedron lies on the lines joning the mid point of the opposite fes Distne of entroid from ll the verties re equl None of these

19 5 The eqution of the line + + = = 0 written in the smmetril form is () () () 0 / / 6 Consider the plnes = 0 nd + = The plne = 0 isets the ngle etween the plnes whih () ontins origin () is ute () is otuse none of these Assertion-Reson Tpe Eh question ontins STATEMENT- (Assertion) nd STATEMENT- (Reson) Eh question hs hoies (A) (B) (C) nd (D) out of whih ONLY ONE is orret (A) Sttement- is True Sttement- is True; Sttement- is orret eplntion for Sttement- (B) Sttement- is True Sttement- is True; Sttement- is NOT orret eplntion for Sttement- (C) Sttement- is True Sttement- is Flse (D) Sttement- is Flse Sttement- is True Consider the plnes 6 = 5 nd + = 5 STATEMENT- : The prmetri equtions of the line of intersetion of the given plnes re = + t = + t = 5t STATEMENT- : The vetor i j 5k is prllel to the line of intersetion of given plnes MD 9 The equtions of two stright lines re nd STATEMENT- : The given lines re oplnr STATEMENT- : The equtions r s = r + s = r + s = 5 re onsistent STATEMENT- : The plnes + = 0 + = 0 nd + = 0 meet in unique point STATEMENT- : The ove plnes hve ommon line of intersetion STATEMENT- : Two plnes + = 6 nd = re prllel STATEMENT- : If the two plnes re prllel then the hve the sme diretion rtions of normls 5 STATEMENT- : e the diretion osines of n direted line STATEMENT- : If l m n re the diretion osines of line then l + m + n = 6 STATEMENT- : There e unique eqution of the plne pssing through the points ( 5) ( ) nd ( ) STATEMENT- : The three given points re olliner STATEMENT- : 0 represents pir of plnes STATEMENT- : The generl eqution of seond degree in ie eqution h + g + f = 0 represents pir of plnes iff + fgh f h = 0 (Answers) EXCERCISE BASED ON NEW PATTERN COMPREHENSION TYPE d 5 6 d d 6 d 8 MATRIX-MATCH TYPE [A-Q B-P C-S D-R] [A-S B-R C-R D-P] [A-R B-S C-P D-Q] [A-R B-P C-Q D-S] MULTIPLE CORRECT CHOICE TYPE d 5 d 6 ASSERTION-REASON TYPE D A D A 5 D 6 A A

20 Plnes re drwn prllel to the o-ordinte plnes through the points ( ) nd ( 5) Find the lengths of the edges of the prllelopiped so formed Show tht the points A( ) B( ) nd C ( 0) re olliner Also find the rtio in whih C divides AB The verties of tringle re A(5 6) B( ) nd C( ) The internl isetor of BAC meets BC in D Find AD Find the points of trisetion of the line segment joining the points ( ) nd (5 5) 5 Find the ngle etween n two digonls of ue 6 If P Q R S re ( 6 ) ( 5 ) (6 ) (0 ) respetivel find the projetion of PQ on RS Find the rtio in whih the surfe + + = 5 divides the line joining (0 ) nd ( 5) 8 Wht re the diretion osines of line tht psses through the points P(6 ) nd Q( ) nd is so direted tht it mkes n ute ngle with the positive diretion of -is 9 If Q e the foot of perpendiulr from P( ) on the line joining the points A( ) nd B( 5) find the o-ordintes of Q 0 Find the projetion of the line joining ( ) nd ( ) on the line hving diretion rtios 6 Find the lous of point whih moves suh tht the sum of its distne from points A(0 0 ) nd B(0 0 ) is onstnt Determine the vlues of nd so tht the points ( ) ( 0 ) nd ( ) re olliner Two verties of tringle re A( ) nd B( ) The medins of the tringle interset t ( ) find the remining verte C of the tringle If the diretion osines of three oplnr lines e l m n ; l m n nd l m n prove tht l l l m m m n n n 0 5 If the diretion osines of two lines t right ngles e l m n nd l m n find the diretion osines of line whih is perpendiulr to oth 6 Find the eqution of the plne through the points ( ) nd ( ) nd prllel to the -is MD 0 INITIAL STEP EXERCISE (SUBJECTIVE) Find the eqution of the plne perpendiulr to the plne + + = 8 nd pssing through the points ( ) nd (9 6) 8 If plne psses through the point ( ) nd is norml to the line joining the points ( 6 ) nd ( 0) find its eqution 9 Find the eqution of the plne pssing through the point ( g) nd perpendiulr to the plnes d = 0 nd d = 0 0 Find the eqution of the plne pssing through the line of intersetion of the plnes 5 = nd + + = 8 nd the point ( ) The plne = is rotted through 90 0 out its line of intersetion with the plne + + = Find its eqution in the new position Find the ngle etween the plne = 0 nd the line whose diretion osines re Find the plne whih isets the otuse ngle etween the plnes + + = 0 nd + + = 9 Find the plnes iseting nd ngles etween plnes + + = 9 nd + + = 0 Whih of these isetor plnes isets the ute ngle etween the given plnes Does origin lie in the ute ngle or otuse ngle etween the given plnes? 5 If vrile plne forms tetrhedron of onstnt volume 6k with the o-ordinte plnes find the lous of the entroid of the tetrhedron 6 Find the lous of point the sum of squres of whose distnes from the plnes = 0 + = 0 nd + + = 0 is 6 A vrile plne t onstnt distne p from the origin meets the o-ordintes es t P Q nd R Find the lous of the point of intersetion of plnes drwn through P Q R nd prllel to the o-ordintes plnes 8 If vrile plne uts the o-ordinte es in A B nd C nd is t onstnt distne p from the origin find the lous of the entroid of the tetrhedron OABC

21 9 Find the eqution of the line drwn through point ( 0 ) to meet t right ngles the line 0 Find the imge of the point P( 5 ) in the plne + + = 6 Find the eqution of the line of intersetion of plnes + 5 = 8 + = Show tht the line represented equtions = + = + d in smmetril form is d Find the point of intersetion of the line + = Show tht the lines nd the plne nd re oplnr Also find the eqution of the plne ontining them 5 Are the lines = 0 = nd oplnr? If es find 5 their point of intersetion nd eqution of the plne in whih the lie 6 Find the eqution of the line whih n e drwn from the point ( 0) to interset the lines orthogonll nd 5 Find the ngle etween the lines = = 0 nd + = = 0 8 Find the o-ordintes of the point where the line + = + = 5 intersets the plne + = 9 Find the length of the shortest distne etween the lines MD 0 The diretion osine of the shortest distne etween the lines re Find (i) its eqution (ii) 5 nd the points where it intersets the lines Find the distne of the point ( 0 ) from the plne = 9 mesured prllel to the line 6 6 Find the o-ordintes of the foot of perpendiulr from the point ( 6 ) to the line Also find the eqution of this perpendiulr Find the eqution of the plne pssing through ( 0) whih ontins the line Find the eqution of the plne through the line l m n l m n nd prllel to the line 5 Let A = ( ) nd B = ( 5) P is vrile point suh tht PAB is n equilterl tringle Find the eqution of the lous of the point P 6 A plne meets the oordintes es t A B nd C suh tht the entroid of the ABC is the point ( ) Find the eqution of the plne ABC Let PM e the perpendiulr from the point P( ) to the - plne If OP mkes n ngle with the positive diretion of the -is nd OM mkes n ngle with the positive diretion of the -is where O is the origin then find nd nd

22 If the points P Q R S re ( 8) ( ) ( ) nd ( 5) respetivel show tht PQ nd RS interset Also find the point of intersetion The diretion osines of three mutull perpendiulr lines re l m n ; l m n nd l m n Prove tht the diretion osines of the line equll inlined to them re l l l m m m n n n The diretion osines of two lines re given the equtions m + n + 5l = 0 6nl lm + 5mn = 0 Find the ngle etween them? Prove tht the line joining the mid-point of opposite edges of tetrhedron re onurrent 5 Prove tht the two lines whose diretion osines re given the reltion pl + qm + rn = 0 nd l + m + n = 0 re perpendiulr if p ( + ) + q ( + ) + r ( + ) = 0 nd prllel if p q r 0 6 If line mkes ngles with the digonls of ue prove tht os os os os If the edges of retngulr prllelopiped re p q r show tht the ngle etween four digonls re given p q r os p q r 8 If two pirs of opposite edges of tetrhedron re mutull perpendiulr show tht the third pir will lso e mutull perpendiulr 9 Find the length nd diretion osines of line segment whose projetions on the o-ordintes es re 6 0 Find the eqution of line of intersetion of the plnes r(i j k) nd r(i j k) MD FINAL STEP EXERCISE (SUBJECTIVE) A tringle is so pled tht the middle points of its sides re on the es If e the length of its sides show tht the eqution to its plne is where 8 = + 8 = + nd 8 = + The plne d + = 0 is rotted through n ngle out its line of intersetion with the plne = 0 Show tht the eqution to the plne in new position is tn 0 Find the refletion of the plne d = 0 in the plne d 0 If P e n point on the plne l + m + n = p nd Q e point on the line OP suh tht OP OQ = p show tht the lous of the point Q is p(l + m + n) = A point P moves on plne A plne through P nd perpendiulr to OP meets the o-ordinte es in A B nd C If the plnes through A B nd C prllel to the plnes = 0 = 0 = 0 interset in Q find the lous of Q 6 P is given point nd PM nd PN re perpendiulr from P to nd -plne respetivel If OP mkes ngles with the plne OMN nd the o-ordintes plnes respetivel prove tht ot = ot + ot + ot + where O is the origin Through point P(h k l) plne is drwn t right ngles to OP to meet the o-ordintes es in A B nd C If OP = p show tht the re of ABC is p 5 hkl 8 Find the volume of the tetrhedron with verties P( ) Q( ) R( ) nd S( ) 9 Find the eqution of the projetion of the line + + = 9 on the plne

23 0 Find the imge of the point ( ) with respet to the plne + + = 6 Hene find the imge of the line plne with respet to the Find the eqution of the line whih psses through the point P( ) nd is prllel to the line d = 0 = d = 0 Find the eqution of the projetion of line + = 0 + = 0 on the plne + + = 0 MD Find the eqution of the plne whih psses through the line d = 0 = d = 0 nd whih is prllel to the line l m n If the plnes = 0 + = 0 nd + = 0 pss through stright line then find the vlue of If the plnes = + = + nd = + meet in line show tht the line of intersetion of these plnes is 6 Prove tht the shortest distne etween n two opposite edges of tetrhedron formed the plnes + = 0 + = 0 + = = is ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE) 6 8 C divides AB eternll in the rtio : 50 unit 8 Q ( 0 ) 5 os 6 units AB internll in the rtio nd eternll in the rtio Q = = C ( 5 5) 5 (m n m n ) (n l n l ) (l m l m ) 6 + = + 5 = = 0 9 ( ) ( ) + ( )( ) + ( ) ( ) = = = os = = = 56 5 = 6k = 6 ( is onstnt quntit)

24 + + = p = 6p MD 9 0 ( 9 ) = 0 5 ( ) ( ) (i) = = 0 (ii) 6 nd = 0 (m n m n ) ( ) + (l n l n ) ( ) + (l m l m ) ( ) = = = tn tn ANSWERS SUBJECTIVE (FINAL STEP EXERCISE) 5 9 A os 6 9 Length = ; 6 ( )( d) ( 5 0 r (06i j k) t( i j k) 8 ½ unit )( d) 9 5 ( d ) ( l + m + n) ( l + m + n) ( d ) = 0 5

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 r

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