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Page 1 Solved Examples on 3-Dimensional Geometry JEE Mains Q1. The points ( ?? , ?? , ?? ) , ( ?? , - ?? , ?? ) and ( ?? , - ?? , ?? ) are collinear, if ?? is equal to (a) -2 (b) 2 (c) 3 (d) -1 Ans: (a) If given points are collinear, then ?? 1 - ?? 2 ?? 2 - ?? 3 = ?? 1 - ?? 2 ?? 2 - ?? 3 = ?? 1 - ?? 2 ?? 2 - ?? 3 ? 5 - 6 6 - 8 = 2 + 1 - 1 + 7 = 4 - 2 2 - ?? ? - 1 - 2 = 3 6 = 2 2 - ?? ? 1 2 = 2 2 - ?? ? ?? = - 2 Q2. A line which makes angle ???? ° with ?? -axis and ?? -axis, then the angle which it makes with ?? -axis is (a) ???? ° (b) ???? ° (c) ???? ° (d) ???? ° Ans: (a) Given that ?? = ?? = 60 ° i.e. ?? = c os ? ?? = c os ? 60 ° = 1 / 2 , ?? = c os ? ?? = c os ? 60 ° = 1 / 2 ? ?? 2 + ?? 2 + ?? 2 = 1 ? ?? 2 = 1 - 1 4 - 1 4 = 1 2 ? ?? = 1 v 2 ? c os ? ?? = 1 v 2 ? ?? = 45 ° Q3. If ?? is a vector of magnitude 21 and has d.r.'s ?? , - ?? , ?? . Then ?? is equal to (a) ???? - ???? + ?????? (b) ???? + ???? + ?????? (c) ???? - ???? - ?????? (d) ???? + ???? - ?????? Ans: (a) D.r.'s of ?? are 2, - 3 , 6. Therefore, its d.c.'s are ?? = 2 7 , ?? = - 3 7 , ?? = 6 7 ? ?? = | ?? | ( ?? + ?? ?? + ?? ?? ) = 21 [ 2 7 ?? - 3 7 ?? + 6 7 ?? ] = 6 ?? - 9 ?? + 18 ?? . Q4. If ?? ?? , ?? ?? , ?? ?? and ?? ?? , ?? ?? , ?? ?? are d.c.'s of two lines inclined to each other at an angle ?? , then the d.c.'s of the internal bisectors of angle between these lines are (a) ?? ?? + ?? ?? ???????? ? ?? / ?? , ?? ?? + ?? ?? ???????? ? ?? / ?? , ?? ?? + ?? ?? ???????? ? ?? / ?? (b) ?? ?? + ?? ?? ???? ???? ? ?? / ?? , ?? ?? + ?? ?? ???? ???? ? ?? / ?? , ?? ?? + ?? ?? ???? ???? ? ?? / ?? (c) ?? ?? - ?? ?? ???????? ? ?? / ?? , ?? ?? - ?? ?? ???????? ? ?? / ?? , ?? ?? - ?? ?? ???????? ? ?? / ?? (d) ?? ?? - ?? ?? ???? ???? ? ?? / ?? , ?? ?? - ?? ?? ???? ???? ? ?? / ?? , ?? ?? - ?? ?? ???? ???? ? ?? / ?? Ans: (b) Let ???? and ???? be two lines. D.c.'s of ???? is ( ?? 1 , ?? 1 , ?? 1 ) and ???? is ( ?? 2 , ?? 2 , ?? 2 ) . Page 2 Solved Examples on 3-Dimensional Geometry JEE Mains Q1. The points ( ?? , ?? , ?? ) , ( ?? , - ?? , ?? ) and ( ?? , - ?? , ?? ) are collinear, if ?? is equal to (a) -2 (b) 2 (c) 3 (d) -1 Ans: (a) If given points are collinear, then ?? 1 - ?? 2 ?? 2 - ?? 3 = ?? 1 - ?? 2 ?? 2 - ?? 3 = ?? 1 - ?? 2 ?? 2 - ?? 3 ? 5 - 6 6 - 8 = 2 + 1 - 1 + 7 = 4 - 2 2 - ?? ? - 1 - 2 = 3 6 = 2 2 - ?? ? 1 2 = 2 2 - ?? ? ?? = - 2 Q2. A line which makes angle ???? ° with ?? -axis and ?? -axis, then the angle which it makes with ?? -axis is (a) ???? ° (b) ???? ° (c) ???? ° (d) ???? ° Ans: (a) Given that ?? = ?? = 60 ° i.e. ?? = c os ? ?? = c os ? 60 ° = 1 / 2 , ?? = c os ? ?? = c os ? 60 ° = 1 / 2 ? ?? 2 + ?? 2 + ?? 2 = 1 ? ?? 2 = 1 - 1 4 - 1 4 = 1 2 ? ?? = 1 v 2 ? c os ? ?? = 1 v 2 ? ?? = 45 ° Q3. If ?? is a vector of magnitude 21 and has d.r.'s ?? , - ?? , ?? . Then ?? is equal to (a) ???? - ???? + ?????? (b) ???? + ???? + ?????? (c) ???? - ???? - ?????? (d) ???? + ???? - ?????? Ans: (a) D.r.'s of ?? are 2, - 3 , 6. Therefore, its d.c.'s are ?? = 2 7 , ?? = - 3 7 , ?? = 6 7 ? ?? = | ?? | ( ?? + ?? ?? + ?? ?? ) = 21 [ 2 7 ?? - 3 7 ?? + 6 7 ?? ] = 6 ?? - 9 ?? + 18 ?? . Q4. If ?? ?? , ?? ?? , ?? ?? and ?? ?? , ?? ?? , ?? ?? are d.c.'s of two lines inclined to each other at an angle ?? , then the d.c.'s of the internal bisectors of angle between these lines are (a) ?? ?? + ?? ?? ???????? ? ?? / ?? , ?? ?? + ?? ?? ???????? ? ?? / ?? , ?? ?? + ?? ?? ???????? ? ?? / ?? (b) ?? ?? + ?? ?? ???? ???? ? ?? / ?? , ?? ?? + ?? ?? ???? ???? ? ?? / ?? , ?? ?? + ?? ?? ???? ???? ? ?? / ?? (c) ?? ?? - ?? ?? ???????? ? ?? / ?? , ?? ?? - ?? ?? ???????? ? ?? / ?? , ?? ?? - ?? ?? ???????? ? ?? / ?? (d) ?? ?? - ?? ?? ???? ???? ? ?? / ?? , ?? ?? - ?? ?? ???? ???? ? ?? / ?? , ?? ?? - ?? ?? ???? ???? ? ?? / ?? Ans: (b) Let ???? and ???? be two lines. D.c.'s of ???? is ( ?? 1 , ?? 1 , ?? 1 ) and ???? is ( ?? 2 , ?? 2 , ?? 2 ) . Let ???? = ???? = 1. Then the co-ordinates of ?? and ?? are ( ?? 1 , ?? 1 , ?? 1 ) and ( ?? 2 , ?? 2 , ?? 2 ) . Let ???? be the bisector of ? ?????? . ( 0 , 0 , 0 ) Then ?? is the mid-point of ???? and so its co-ordinates are ( ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 ). ? D.r.'s of line ???? are ( ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 ) We have, ???? ? = v ( ?? 1 + ?? 2 2 ) 2 + ( ?? 1 + ?? 2 2 ) 2 + ( ?? 1 + ?? 2 2 ) 2 = 1 2 v ?? 1 2 + ?? 1 2 + ?? 1 2 + ?? 2 2 + ?? 2 2 + ?? 2 2 + 2 ( ?? 1 ?? 2 + ?? 1 ?? 2 + ?? 1 ?? 2 ) ? = 1 2 v 1 + 1 + 2 co s ? ?? = 1 2 v 2 ( 2 · co s 2 ? ?? / 2 ) = co s ? ?? / 2 . D.r.'s of line ???? are ?? 1 + ?? 2 2 c os ? ?? / 2 , ?? 1 + ?? 2 2 c os ? ?? / 2 , ?? 1 + ?? 2 2 c os ? ?? / 2 . Q5. The equation to the straight line passing through the points ( ?? , - ?? , - ?? ) and ( - ?? , ?? , ?? ) is (a) ?? - ?? ?? = ?? + ?? - ?? = ?? + ?? - ?? (b) ?? + ?? ?? = ?? - ?? ?? = ?? - ?? - ?? (c) ?? - ?? = ?? ?? = ?? ?? (d) ?? ?? = ?? - ?? = ?? - ?? Ans: (a) We know that equation of a straight line is of the form ?? - ?? 1 ?? = ?? - ?? 1 ?? = ?? - ?? 1 ?? D.r.'s of the line = ( - 1 - 4 , 5 + 5 , 3 + 2 ) i.e., ( - 5 , 10 , 5 ) or ( - 1 , 2 , 1 ) . Hence the equation is ?? - 4 - 1 = ?? + 5 2 = ?? + 2 1 i.e., ?? - 4 1 = ?? + 5 - 2 = ?? + 2 - 1 Q6. The direction ratio of the line which is perpendicular to the lines ?? - ?? ?? = ?? + ???? - ?? = ?? - ?? ?? and ?? + ?? ?? = ?? + ?? ?? = ?? - ?? - ?? are (a) ? ?? , ?? , ?? ? (b) ? ?? , - ?? , ?? ? (c) ? ?? , - ?? , - ?? ? (d) ? - ?? , ?? , ?? ? Ans: (a) Let d.r.'s of line be ?? , ?? , ?? . ? ? line is perpendicul ? ? 2 ?? - 3 ?? + ?? = 0 ?? + 2 ?? - 2 ?? = 0 From equation (i) and (ii) ?? 6 - 2 = ?? 1 + 4 = ?? 4 + 3 or ?? 4 = ?? 5 = ?? 7 . Hence, d.r.'s of line ( < 4 , 5 , 7 ? ) Q7. If the line ?? - ?? ?? = ?? + ?? ?? = ?? - ?? ?? and ?? - ?? ?? = ?? - ?? ?? = ?? ?? intersect, then ?? = (a) ?? / ?? (b) ?? / ?? Page 3 Solved Examples on 3-Dimensional Geometry JEE Mains Q1. The points ( ?? , ?? , ?? ) , ( ?? , - ?? , ?? ) and ( ?? , - ?? , ?? ) are collinear, if ?? is equal to (a) -2 (b) 2 (c) 3 (d) -1 Ans: (a) If given points are collinear, then ?? 1 - ?? 2 ?? 2 - ?? 3 = ?? 1 - ?? 2 ?? 2 - ?? 3 = ?? 1 - ?? 2 ?? 2 - ?? 3 ? 5 - 6 6 - 8 = 2 + 1 - 1 + 7 = 4 - 2 2 - ?? ? - 1 - 2 = 3 6 = 2 2 - ?? ? 1 2 = 2 2 - ?? ? ?? = - 2 Q2. A line which makes angle ???? ° with ?? -axis and ?? -axis, then the angle which it makes with ?? -axis is (a) ???? ° (b) ???? ° (c) ???? ° (d) ???? ° Ans: (a) Given that ?? = ?? = 60 ° i.e. ?? = c os ? ?? = c os ? 60 ° = 1 / 2 , ?? = c os ? ?? = c os ? 60 ° = 1 / 2 ? ?? 2 + ?? 2 + ?? 2 = 1 ? ?? 2 = 1 - 1 4 - 1 4 = 1 2 ? ?? = 1 v 2 ? c os ? ?? = 1 v 2 ? ?? = 45 ° Q3. If ?? is a vector of magnitude 21 and has d.r.'s ?? , - ?? , ?? . Then ?? is equal to (a) ???? - ???? + ?????? (b) ???? + ???? + ?????? (c) ???? - ???? - ?????? (d) ???? + ???? - ?????? Ans: (a) D.r.'s of ?? are 2, - 3 , 6. Therefore, its d.c.'s are ?? = 2 7 , ?? = - 3 7 , ?? = 6 7 ? ?? = | ?? | ( ?? + ?? ?? + ?? ?? ) = 21 [ 2 7 ?? - 3 7 ?? + 6 7 ?? ] = 6 ?? - 9 ?? + 18 ?? . Q4. If ?? ?? , ?? ?? , ?? ?? and ?? ?? , ?? ?? , ?? ?? are d.c.'s of two lines inclined to each other at an angle ?? , then the d.c.'s of the internal bisectors of angle between these lines are (a) ?? ?? + ?? ?? ???????? ? ?? / ?? , ?? ?? + ?? ?? ???????? ? ?? / ?? , ?? ?? + ?? ?? ???????? ? ?? / ?? (b) ?? ?? + ?? ?? ???? ???? ? ?? / ?? , ?? ?? + ?? ?? ???? ???? ? ?? / ?? , ?? ?? + ?? ?? ???? ???? ? ?? / ?? (c) ?? ?? - ?? ?? ???????? ? ?? / ?? , ?? ?? - ?? ?? ???????? ? ?? / ?? , ?? ?? - ?? ?? ???????? ? ?? / ?? (d) ?? ?? - ?? ?? ???? ???? ? ?? / ?? , ?? ?? - ?? ?? ???? ???? ? ?? / ?? , ?? ?? - ?? ?? ???? ???? ? ?? / ?? Ans: (b) Let ???? and ???? be two lines. D.c.'s of ???? is ( ?? 1 , ?? 1 , ?? 1 ) and ???? is ( ?? 2 , ?? 2 , ?? 2 ) . Let ???? = ???? = 1. Then the co-ordinates of ?? and ?? are ( ?? 1 , ?? 1 , ?? 1 ) and ( ?? 2 , ?? 2 , ?? 2 ) . Let ???? be the bisector of ? ?????? . ( 0 , 0 , 0 ) Then ?? is the mid-point of ???? and so its co-ordinates are ( ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 ). ? D.r.'s of line ???? are ( ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 ) We have, ???? ? = v ( ?? 1 + ?? 2 2 ) 2 + ( ?? 1 + ?? 2 2 ) 2 + ( ?? 1 + ?? 2 2 ) 2 = 1 2 v ?? 1 2 + ?? 1 2 + ?? 1 2 + ?? 2 2 + ?? 2 2 + ?? 2 2 + 2 ( ?? 1 ?? 2 + ?? 1 ?? 2 + ?? 1 ?? 2 ) ? = 1 2 v 1 + 1 + 2 co s ? ?? = 1 2 v 2 ( 2 · co s 2 ? ?? / 2 ) = co s ? ?? / 2 . D.r.'s of line ???? are ?? 1 + ?? 2 2 c os ? ?? / 2 , ?? 1 + ?? 2 2 c os ? ?? / 2 , ?? 1 + ?? 2 2 c os ? ?? / 2 . Q5. The equation to the straight line passing through the points ( ?? , - ?? , - ?? ) and ( - ?? , ?? , ?? ) is (a) ?? - ?? ?? = ?? + ?? - ?? = ?? + ?? - ?? (b) ?? + ?? ?? = ?? - ?? ?? = ?? - ?? - ?? (c) ?? - ?? = ?? ?? = ?? ?? (d) ?? ?? = ?? - ?? = ?? - ?? Ans: (a) We know that equation of a straight line is of the form ?? - ?? 1 ?? = ?? - ?? 1 ?? = ?? - ?? 1 ?? D.r.'s of the line = ( - 1 - 4 , 5 + 5 , 3 + 2 ) i.e., ( - 5 , 10 , 5 ) or ( - 1 , 2 , 1 ) . Hence the equation is ?? - 4 - 1 = ?? + 5 2 = ?? + 2 1 i.e., ?? - 4 1 = ?? + 5 - 2 = ?? + 2 - 1 Q6. The direction ratio of the line which is perpendicular to the lines ?? - ?? ?? = ?? + ???? - ?? = ?? - ?? ?? and ?? + ?? ?? = ?? + ?? ?? = ?? - ?? - ?? are (a) ? ?? , ?? , ?? ? (b) ? ?? , - ?? , ?? ? (c) ? ?? , - ?? , - ?? ? (d) ? - ?? , ?? , ?? ? Ans: (a) Let d.r.'s of line be ?? , ?? , ?? . ? ? line is perpendicul ? ? 2 ?? - 3 ?? + ?? = 0 ?? + 2 ?? - 2 ?? = 0 From equation (i) and (ii) ?? 6 - 2 = ?? 1 + 4 = ?? 4 + 3 or ?? 4 = ?? 5 = ?? 7 . Hence, d.r.'s of line ( < 4 , 5 , 7 ? ) Q7. If the line ?? - ?? ?? = ?? + ?? ?? = ?? - ?? ?? and ?? - ?? ?? = ?? - ?? ?? = ?? ?? intersect, then ?? = (a) ?? / ?? (b) ?? / ?? (c) 0 (d) -1 Ans: (b) We have, ?? - 1 2 = ?? + 1 3 = ?? - 1 4 = ?? 1 (Let) ?? = 2 ?? 1 + 1 , ?? = 3 ?? 1 - 1 , ?? = 4 ?? 1 + 1 i.e. point is ( 2 ?? 1 + 1 , 3 ?? 1 - 1 , 4 ?? 1 + 1 ) and ?? - 3 1 = ?? - ?? 2 = ?? 1 = ?? 2 (Let) i.e. point is ( ?? 2 + 3 , 2 ?? 2 + ?? , ?? 2 ) . If the lines are intersecting, then they have a common point. ? 2 ?? 1 + 1 = ?? 2 + 3 , 3 ?? 1 - 1 = 2 ?? 2 + ?? , 4 ?? 1 + 1 = ?? 2 On solving, ?? 1 = - 3 / 2 , ?? 2 = - 5 Hence, ?? = 9 / 2. Q8. The length of the perpendicular from the origin to line ?? = ( ???? + ???? + ???? ) + ?? ( ???? + ???? - ???? ) is (a) ?? v ?? (b) 2 (c) ?? v ?? (d) 6 Ans: (d) ?? ? = 0 . ?? + 0 . ?? + 0 . ?? ???? ? ? ? ? ? = ( ?? - ?? ? ) - ( ( ?? - ?? ? ) · ?? | ?? | 2 ) ?? ???? ? ? ? ? ? = ( 4 ?? + 2 ?? + 4 ?? ) - [ ( 4 ?? + 2 ?? + 4 ?? ) · ( 3 ?? + 4 ?? - 5 ?? ) 9 + 16 + 25 ] ( 3 ?? + 4 ?? - 5 ?? ) = 4 ?? + 2 ?? + 4 ?? - ( 12 + 8 - 20 50 ) · ( 3 ?? + 4 ?? - 5 ?? ) ???? ? ? ? ? ? = 4 ?? + 2 ?? + 4 ?? The length of ???? is magnitude of ???? ? ? ? ? ? i.e., Length of perpendicular = | ???? ? ? ? ? ? | = v 16 + 4 + 16 = 6. Q9. If the straight lines ?? = ?? + ?? , ?? = ?? - ???? , ?? = ?? + ???? and ?? = ?? ?? , ?? = ?? + ?? , ?? = ?? - ?? , with parameters ?? and ?? respectively, are coplanar, then ?? equals (a) 0 (b) -1 (c) - ?? ?? (d) -2 Ans: (d) We have ?? - 1 1 = ?? + 3 - ?? = ?? - 1 ?? = ?? and 2 ?? 1 = ?? - 1 1 = ?? - 2 - 1 = ?? i.e. ?? - 0 1 = ?? - 1 2 = ?? - 2 - 2 = ?? 2 Since, lines are co-planar, Page 4 Solved Examples on 3-Dimensional Geometry JEE Mains Q1. The points ( ?? , ?? , ?? ) , ( ?? , - ?? , ?? ) and ( ?? , - ?? , ?? ) are collinear, if ?? is equal to (a) -2 (b) 2 (c) 3 (d) -1 Ans: (a) If given points are collinear, then ?? 1 - ?? 2 ?? 2 - ?? 3 = ?? 1 - ?? 2 ?? 2 - ?? 3 = ?? 1 - ?? 2 ?? 2 - ?? 3 ? 5 - 6 6 - 8 = 2 + 1 - 1 + 7 = 4 - 2 2 - ?? ? - 1 - 2 = 3 6 = 2 2 - ?? ? 1 2 = 2 2 - ?? ? ?? = - 2 Q2. A line which makes angle ???? ° with ?? -axis and ?? -axis, then the angle which it makes with ?? -axis is (a) ???? ° (b) ???? ° (c) ???? ° (d) ???? ° Ans: (a) Given that ?? = ?? = 60 ° i.e. ?? = c os ? ?? = c os ? 60 ° = 1 / 2 , ?? = c os ? ?? = c os ? 60 ° = 1 / 2 ? ?? 2 + ?? 2 + ?? 2 = 1 ? ?? 2 = 1 - 1 4 - 1 4 = 1 2 ? ?? = 1 v 2 ? c os ? ?? = 1 v 2 ? ?? = 45 ° Q3. If ?? is a vector of magnitude 21 and has d.r.'s ?? , - ?? , ?? . Then ?? is equal to (a) ???? - ???? + ?????? (b) ???? + ???? + ?????? (c) ???? - ???? - ?????? (d) ???? + ???? - ?????? Ans: (a) D.r.'s of ?? are 2, - 3 , 6. Therefore, its d.c.'s are ?? = 2 7 , ?? = - 3 7 , ?? = 6 7 ? ?? = | ?? | ( ?? + ?? ?? + ?? ?? ) = 21 [ 2 7 ?? - 3 7 ?? + 6 7 ?? ] = 6 ?? - 9 ?? + 18 ?? . Q4. If ?? ?? , ?? ?? , ?? ?? and ?? ?? , ?? ?? , ?? ?? are d.c.'s of two lines inclined to each other at an angle ?? , then the d.c.'s of the internal bisectors of angle between these lines are (a) ?? ?? + ?? ?? ???????? ? ?? / ?? , ?? ?? + ?? ?? ???????? ? ?? / ?? , ?? ?? + ?? ?? ???????? ? ?? / ?? (b) ?? ?? + ?? ?? ???? ???? ? ?? / ?? , ?? ?? + ?? ?? ???? ???? ? ?? / ?? , ?? ?? + ?? ?? ???? ???? ? ?? / ?? (c) ?? ?? - ?? ?? ???????? ? ?? / ?? , ?? ?? - ?? ?? ???????? ? ?? / ?? , ?? ?? - ?? ?? ???????? ? ?? / ?? (d) ?? ?? - ?? ?? ???? ???? ? ?? / ?? , ?? ?? - ?? ?? ???? ???? ? ?? / ?? , ?? ?? - ?? ?? ???? ???? ? ?? / ?? Ans: (b) Let ???? and ???? be two lines. D.c.'s of ???? is ( ?? 1 , ?? 1 , ?? 1 ) and ???? is ( ?? 2 , ?? 2 , ?? 2 ) . Let ???? = ???? = 1. Then the co-ordinates of ?? and ?? are ( ?? 1 , ?? 1 , ?? 1 ) and ( ?? 2 , ?? 2 , ?? 2 ) . Let ???? be the bisector of ? ?????? . ( 0 , 0 , 0 ) Then ?? is the mid-point of ???? and so its co-ordinates are ( ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 ). ? D.r.'s of line ???? are ( ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 ) We have, ???? ? = v ( ?? 1 + ?? 2 2 ) 2 + ( ?? 1 + ?? 2 2 ) 2 + ( ?? 1 + ?? 2 2 ) 2 = 1 2 v ?? 1 2 + ?? 1 2 + ?? 1 2 + ?? 2 2 + ?? 2 2 + ?? 2 2 + 2 ( ?? 1 ?? 2 + ?? 1 ?? 2 + ?? 1 ?? 2 ) ? = 1 2 v 1 + 1 + 2 co s ? ?? = 1 2 v 2 ( 2 · co s 2 ? ?? / 2 ) = co s ? ?? / 2 . D.r.'s of line ???? are ?? 1 + ?? 2 2 c os ? ?? / 2 , ?? 1 + ?? 2 2 c os ? ?? / 2 , ?? 1 + ?? 2 2 c os ? ?? / 2 . Q5. The equation to the straight line passing through the points ( ?? , - ?? , - ?? ) and ( - ?? , ?? , ?? ) is (a) ?? - ?? ?? = ?? + ?? - ?? = ?? + ?? - ?? (b) ?? + ?? ?? = ?? - ?? ?? = ?? - ?? - ?? (c) ?? - ?? = ?? ?? = ?? ?? (d) ?? ?? = ?? - ?? = ?? - ?? Ans: (a) We know that equation of a straight line is of the form ?? - ?? 1 ?? = ?? - ?? 1 ?? = ?? - ?? 1 ?? D.r.'s of the line = ( - 1 - 4 , 5 + 5 , 3 + 2 ) i.e., ( - 5 , 10 , 5 ) or ( - 1 , 2 , 1 ) . Hence the equation is ?? - 4 - 1 = ?? + 5 2 = ?? + 2 1 i.e., ?? - 4 1 = ?? + 5 - 2 = ?? + 2 - 1 Q6. The direction ratio of the line which is perpendicular to the lines ?? - ?? ?? = ?? + ???? - ?? = ?? - ?? ?? and ?? + ?? ?? = ?? + ?? ?? = ?? - ?? - ?? are (a) ? ?? , ?? , ?? ? (b) ? ?? , - ?? , ?? ? (c) ? ?? , - ?? , - ?? ? (d) ? - ?? , ?? , ?? ? Ans: (a) Let d.r.'s of line be ?? , ?? , ?? . ? ? line is perpendicul ? ? 2 ?? - 3 ?? + ?? = 0 ?? + 2 ?? - 2 ?? = 0 From equation (i) and (ii) ?? 6 - 2 = ?? 1 + 4 = ?? 4 + 3 or ?? 4 = ?? 5 = ?? 7 . Hence, d.r.'s of line ( < 4 , 5 , 7 ? ) Q7. If the line ?? - ?? ?? = ?? + ?? ?? = ?? - ?? ?? and ?? - ?? ?? = ?? - ?? ?? = ?? ?? intersect, then ?? = (a) ?? / ?? (b) ?? / ?? (c) 0 (d) -1 Ans: (b) We have, ?? - 1 2 = ?? + 1 3 = ?? - 1 4 = ?? 1 (Let) ?? = 2 ?? 1 + 1 , ?? = 3 ?? 1 - 1 , ?? = 4 ?? 1 + 1 i.e. point is ( 2 ?? 1 + 1 , 3 ?? 1 - 1 , 4 ?? 1 + 1 ) and ?? - 3 1 = ?? - ?? 2 = ?? 1 = ?? 2 (Let) i.e. point is ( ?? 2 + 3 , 2 ?? 2 + ?? , ?? 2 ) . If the lines are intersecting, then they have a common point. ? 2 ?? 1 + 1 = ?? 2 + 3 , 3 ?? 1 - 1 = 2 ?? 2 + ?? , 4 ?? 1 + 1 = ?? 2 On solving, ?? 1 = - 3 / 2 , ?? 2 = - 5 Hence, ?? = 9 / 2. Q8. The length of the perpendicular from the origin to line ?? = ( ???? + ???? + ???? ) + ?? ( ???? + ???? - ???? ) is (a) ?? v ?? (b) 2 (c) ?? v ?? (d) 6 Ans: (d) ?? ? = 0 . ?? + 0 . ?? + 0 . ?? ???? ? ? ? ? ? = ( ?? - ?? ? ) - ( ( ?? - ?? ? ) · ?? | ?? | 2 ) ?? ???? ? ? ? ? ? = ( 4 ?? + 2 ?? + 4 ?? ) - [ ( 4 ?? + 2 ?? + 4 ?? ) · ( 3 ?? + 4 ?? - 5 ?? ) 9 + 16 + 25 ] ( 3 ?? + 4 ?? - 5 ?? ) = 4 ?? + 2 ?? + 4 ?? - ( 12 + 8 - 20 50 ) · ( 3 ?? + 4 ?? - 5 ?? ) ???? ? ? ? ? ? = 4 ?? + 2 ?? + 4 ?? The length of ???? is magnitude of ???? ? ? ? ? ? i.e., Length of perpendicular = | ???? ? ? ? ? ? | = v 16 + 4 + 16 = 6. Q9. If the straight lines ?? = ?? + ?? , ?? = ?? - ???? , ?? = ?? + ???? and ?? = ?? ?? , ?? = ?? + ?? , ?? = ?? - ?? , with parameters ?? and ?? respectively, are coplanar, then ?? equals (a) 0 (b) -1 (c) - ?? ?? (d) -2 Ans: (d) We have ?? - 1 1 = ?? + 3 - ?? = ?? - 1 ?? = ?? and 2 ?? 1 = ?? - 1 1 = ?? - 2 - 1 = ?? i.e. ?? - 0 1 = ?? - 1 2 = ?? - 2 - 2 = ?? 2 Since, lines are co-planar, Then, | ?? 2 - ?? 1 ?? 2 - ?? 1 ?? 2 - ?? 1 ?? 1 ?? 1 ?? 1 ?? 2 ?? 2 ?? 2 | = 0 ? | - 1 4 1 1 - ?? ?? 1 2 - 2 | = 0 On solving, ?? = - 2. Q10. The equation of plane passing through the points ( ?? , ?? , ?? ) and ( ?? , ?? , ?? ) and perpendicular to the plane ?? ?? + ?? ?? + ?? ?? = ?? is (a) ?? ?? + ?? ?? + ?? ?? = ?? (b) ?? ?? + ?? ?? + ?? ?? = ?? (c) ?? ?? + ?? ?? - ?? ?? = ?? (d) None of these Ans: (c) We know that, equation of plane is ?? ( ?? - ?? 1 ) + ?? ( ?? - ?? 1 ) + ?? ( ?? - ?? 1 ) = 0 It passes through ( 2 , 2 , 1 ) ? ?? ( ?? - 2 ) + ?? ( ?? - 2 ) + ?? ( ?? - 1 ) = 0 Plane (i) also passes through ( 9 , 3 , 6 ) and is perpendicular to the plane 2 ?? + 6 ?? + 6 ?? = 1 ? 7 ?? + ?? + 5 ?? = 0 and 2 ?? + 6 ?? + 6 ?? = 0 ?? 6 - 30 = ?? 10 - 42 = ?? 42 - 2 or ?? - 24 = ?? - 32 = ?? 40 Q11. Angle between two planes ?? + ?? ?? + ?? ?? = ?? and - ?? ?? + ?? ?? + ?? ?? = ?? is (a) ?? ???? - ?? ? ?? v ?? ???? (b) ?? ???? - ?? ? ???? v ?? ???? (c) ?? ???? - ?? ? ?? v ?? ???? (d) ?? ???? - ?? ? ?? v ?? ?? Ans: (a) We know that, c os ? ?? = ?? 1 ?? 2 + ?? 1 ?? 2 + ?? 1 ?? 2 v ?? 1 2 + ?? 1 2 + ?? 1 2 v ?? 2 2 + ?? 2 2 + ?? 2 2 = 1 ( - 5 ) + 2 ( 3 ) + 2 ( 4 ) v 1 + 4 + 4 v 25 + 9 + 16 = 9 3 . 5 v 2 = 3 v 2 10 i.e. ?? = c os - 1 ? ( 3 v 2 10 ). Q12. The distance of the point ( ?? , ?? , - ?? ) from the plane ?? - ?? ?? + ?? ?? = ?? is (a) v ???? ???? (b) ???? ???? (c) ???? v ???? (d) v ???? ???? Ans: (c) Distance of the plane from ( 2 , 1 , - 1 ) = | 2 - 2 ( 1 ) + 4 ( - 1 ) - 9 v 1 + 4 + 16 | = 13 v 21 . Q13. A unit vector perpendicular to plane determined by the points ?? ( ?? , - ?? , ?? ) , ?? ( ?? , ?? , - ?? ) and ?? ( ?? , ?? , ?? ) is (a) ???? - ?? + ?? v ?? Page 5 Solved Examples on 3-Dimensional Geometry JEE Mains Q1. The points ( ?? , ?? , ?? ) , ( ?? , - ?? , ?? ) and ( ?? , - ?? , ?? ) are collinear, if ?? is equal to (a) -2 (b) 2 (c) 3 (d) -1 Ans: (a) If given points are collinear, then ?? 1 - ?? 2 ?? 2 - ?? 3 = ?? 1 - ?? 2 ?? 2 - ?? 3 = ?? 1 - ?? 2 ?? 2 - ?? 3 ? 5 - 6 6 - 8 = 2 + 1 - 1 + 7 = 4 - 2 2 - ?? ? - 1 - 2 = 3 6 = 2 2 - ?? ? 1 2 = 2 2 - ?? ? ?? = - 2 Q2. A line which makes angle ???? ° with ?? -axis and ?? -axis, then the angle which it makes with ?? -axis is (a) ???? ° (b) ???? ° (c) ???? ° (d) ???? ° Ans: (a) Given that ?? = ?? = 60 ° i.e. ?? = c os ? ?? = c os ? 60 ° = 1 / 2 , ?? = c os ? ?? = c os ? 60 ° = 1 / 2 ? ?? 2 + ?? 2 + ?? 2 = 1 ? ?? 2 = 1 - 1 4 - 1 4 = 1 2 ? ?? = 1 v 2 ? c os ? ?? = 1 v 2 ? ?? = 45 ° Q3. If ?? is a vector of magnitude 21 and has d.r.'s ?? , - ?? , ?? . Then ?? is equal to (a) ???? - ???? + ?????? (b) ???? + ???? + ?????? (c) ???? - ???? - ?????? (d) ???? + ???? - ?????? Ans: (a) D.r.'s of ?? are 2, - 3 , 6. Therefore, its d.c.'s are ?? = 2 7 , ?? = - 3 7 , ?? = 6 7 ? ?? = | ?? | ( ?? + ?? ?? + ?? ?? ) = 21 [ 2 7 ?? - 3 7 ?? + 6 7 ?? ] = 6 ?? - 9 ?? + 18 ?? . Q4. If ?? ?? , ?? ?? , ?? ?? and ?? ?? , ?? ?? , ?? ?? are d.c.'s of two lines inclined to each other at an angle ?? , then the d.c.'s of the internal bisectors of angle between these lines are (a) ?? ?? + ?? ?? ???????? ? ?? / ?? , ?? ?? + ?? ?? ???????? ? ?? / ?? , ?? ?? + ?? ?? ???????? ? ?? / ?? (b) ?? ?? + ?? ?? ???? ???? ? ?? / ?? , ?? ?? + ?? ?? ???? ???? ? ?? / ?? , ?? ?? + ?? ?? ???? ???? ? ?? / ?? (c) ?? ?? - ?? ?? ???????? ? ?? / ?? , ?? ?? - ?? ?? ???????? ? ?? / ?? , ?? ?? - ?? ?? ???????? ? ?? / ?? (d) ?? ?? - ?? ?? ???? ???? ? ?? / ?? , ?? ?? - ?? ?? ???? ???? ? ?? / ?? , ?? ?? - ?? ?? ???? ???? ? ?? / ?? Ans: (b) Let ???? and ???? be two lines. D.c.'s of ???? is ( ?? 1 , ?? 1 , ?? 1 ) and ???? is ( ?? 2 , ?? 2 , ?? 2 ) . Let ???? = ???? = 1. Then the co-ordinates of ?? and ?? are ( ?? 1 , ?? 1 , ?? 1 ) and ( ?? 2 , ?? 2 , ?? 2 ) . Let ???? be the bisector of ? ?????? . ( 0 , 0 , 0 ) Then ?? is the mid-point of ???? and so its co-ordinates are ( ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 ). ? D.r.'s of line ???? are ( ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 , ?? 1 + ?? 2 2 ) We have, ???? ? = v ( ?? 1 + ?? 2 2 ) 2 + ( ?? 1 + ?? 2 2 ) 2 + ( ?? 1 + ?? 2 2 ) 2 = 1 2 v ?? 1 2 + ?? 1 2 + ?? 1 2 + ?? 2 2 + ?? 2 2 + ?? 2 2 + 2 ( ?? 1 ?? 2 + ?? 1 ?? 2 + ?? 1 ?? 2 ) ? = 1 2 v 1 + 1 + 2 co s ? ?? = 1 2 v 2 ( 2 · co s 2 ? ?? / 2 ) = co s ? ?? / 2 . D.r.'s of line ???? are ?? 1 + ?? 2 2 c os ? ?? / 2 , ?? 1 + ?? 2 2 c os ? ?? / 2 , ?? 1 + ?? 2 2 c os ? ?? / 2 . Q5. The equation to the straight line passing through the points ( ?? , - ?? , - ?? ) and ( - ?? , ?? , ?? ) is (a) ?? - ?? ?? = ?? + ?? - ?? = ?? + ?? - ?? (b) ?? + ?? ?? = ?? - ?? ?? = ?? - ?? - ?? (c) ?? - ?? = ?? ?? = ?? ?? (d) ?? ?? = ?? - ?? = ?? - ?? Ans: (a) We know that equation of a straight line is of the form ?? - ?? 1 ?? = ?? - ?? 1 ?? = ?? - ?? 1 ?? D.r.'s of the line = ( - 1 - 4 , 5 + 5 , 3 + 2 ) i.e., ( - 5 , 10 , 5 ) or ( - 1 , 2 , 1 ) . Hence the equation is ?? - 4 - 1 = ?? + 5 2 = ?? + 2 1 i.e., ?? - 4 1 = ?? + 5 - 2 = ?? + 2 - 1 Q6. The direction ratio of the line which is perpendicular to the lines ?? - ?? ?? = ?? + ???? - ?? = ?? - ?? ?? and ?? + ?? ?? = ?? + ?? ?? = ?? - ?? - ?? are (a) ? ?? , ?? , ?? ? (b) ? ?? , - ?? , ?? ? (c) ? ?? , - ?? , - ?? ? (d) ? - ?? , ?? , ?? ? Ans: (a) Let d.r.'s of line be ?? , ?? , ?? . ? ? line is perpendicul ? ? 2 ?? - 3 ?? + ?? = 0 ?? + 2 ?? - 2 ?? = 0 From equation (i) and (ii) ?? 6 - 2 = ?? 1 + 4 = ?? 4 + 3 or ?? 4 = ?? 5 = ?? 7 . Hence, d.r.'s of line ( < 4 , 5 , 7 ? ) Q7. If the line ?? - ?? ?? = ?? + ?? ?? = ?? - ?? ?? and ?? - ?? ?? = ?? - ?? ?? = ?? ?? intersect, then ?? = (a) ?? / ?? (b) ?? / ?? (c) 0 (d) -1 Ans: (b) We have, ?? - 1 2 = ?? + 1 3 = ?? - 1 4 = ?? 1 (Let) ?? = 2 ?? 1 + 1 , ?? = 3 ?? 1 - 1 , ?? = 4 ?? 1 + 1 i.e. point is ( 2 ?? 1 + 1 , 3 ?? 1 - 1 , 4 ?? 1 + 1 ) and ?? - 3 1 = ?? - ?? 2 = ?? 1 = ?? 2 (Let) i.e. point is ( ?? 2 + 3 , 2 ?? 2 + ?? , ?? 2 ) . If the lines are intersecting, then they have a common point. ? 2 ?? 1 + 1 = ?? 2 + 3 , 3 ?? 1 - 1 = 2 ?? 2 + ?? , 4 ?? 1 + 1 = ?? 2 On solving, ?? 1 = - 3 / 2 , ?? 2 = - 5 Hence, ?? = 9 / 2. Q8. The length of the perpendicular from the origin to line ?? = ( ???? + ???? + ???? ) + ?? ( ???? + ???? - ???? ) is (a) ?? v ?? (b) 2 (c) ?? v ?? (d) 6 Ans: (d) ?? ? = 0 . ?? + 0 . ?? + 0 . ?? ???? ? ? ? ? ? = ( ?? - ?? ? ) - ( ( ?? - ?? ? ) · ?? | ?? | 2 ) ?? ???? ? ? ? ? ? = ( 4 ?? + 2 ?? + 4 ?? ) - [ ( 4 ?? + 2 ?? + 4 ?? ) · ( 3 ?? + 4 ?? - 5 ?? ) 9 + 16 + 25 ] ( 3 ?? + 4 ?? - 5 ?? ) = 4 ?? + 2 ?? + 4 ?? - ( 12 + 8 - 20 50 ) · ( 3 ?? + 4 ?? - 5 ?? ) ???? ? ? ? ? ? = 4 ?? + 2 ?? + 4 ?? The length of ???? is magnitude of ???? ? ? ? ? ? i.e., Length of perpendicular = | ???? ? ? ? ? ? | = v 16 + 4 + 16 = 6. Q9. If the straight lines ?? = ?? + ?? , ?? = ?? - ???? , ?? = ?? + ???? and ?? = ?? ?? , ?? = ?? + ?? , ?? = ?? - ?? , with parameters ?? and ?? respectively, are coplanar, then ?? equals (a) 0 (b) -1 (c) - ?? ?? (d) -2 Ans: (d) We have ?? - 1 1 = ?? + 3 - ?? = ?? - 1 ?? = ?? and 2 ?? 1 = ?? - 1 1 = ?? - 2 - 1 = ?? i.e. ?? - 0 1 = ?? - 1 2 = ?? - 2 - 2 = ?? 2 Since, lines are co-planar, Then, | ?? 2 - ?? 1 ?? 2 - ?? 1 ?? 2 - ?? 1 ?? 1 ?? 1 ?? 1 ?? 2 ?? 2 ?? 2 | = 0 ? | - 1 4 1 1 - ?? ?? 1 2 - 2 | = 0 On solving, ?? = - 2. Q10. The equation of plane passing through the points ( ?? , ?? , ?? ) and ( ?? , ?? , ?? ) and perpendicular to the plane ?? ?? + ?? ?? + ?? ?? = ?? is (a) ?? ?? + ?? ?? + ?? ?? = ?? (b) ?? ?? + ?? ?? + ?? ?? = ?? (c) ?? ?? + ?? ?? - ?? ?? = ?? (d) None of these Ans: (c) We know that, equation of plane is ?? ( ?? - ?? 1 ) + ?? ( ?? - ?? 1 ) + ?? ( ?? - ?? 1 ) = 0 It passes through ( 2 , 2 , 1 ) ? ?? ( ?? - 2 ) + ?? ( ?? - 2 ) + ?? ( ?? - 1 ) = 0 Plane (i) also passes through ( 9 , 3 , 6 ) and is perpendicular to the plane 2 ?? + 6 ?? + 6 ?? = 1 ? 7 ?? + ?? + 5 ?? = 0 and 2 ?? + 6 ?? + 6 ?? = 0 ?? 6 - 30 = ?? 10 - 42 = ?? 42 - 2 or ?? - 24 = ?? - 32 = ?? 40 Q11. Angle between two planes ?? + ?? ?? + ?? ?? = ?? and - ?? ?? + ?? ?? + ?? ?? = ?? is (a) ?? ???? - ?? ? ?? v ?? ???? (b) ?? ???? - ?? ? ???? v ?? ???? (c) ?? ???? - ?? ? ?? v ?? ???? (d) ?? ???? - ?? ? ?? v ?? ?? Ans: (a) We know that, c os ? ?? = ?? 1 ?? 2 + ?? 1 ?? 2 + ?? 1 ?? 2 v ?? 1 2 + ?? 1 2 + ?? 1 2 v ?? 2 2 + ?? 2 2 + ?? 2 2 = 1 ( - 5 ) + 2 ( 3 ) + 2 ( 4 ) v 1 + 4 + 4 v 25 + 9 + 16 = 9 3 . 5 v 2 = 3 v 2 10 i.e. ?? = c os - 1 ? ( 3 v 2 10 ). Q12. The distance of the point ( ?? , ?? , - ?? ) from the plane ?? - ?? ?? + ?? ?? = ?? is (a) v ???? ???? (b) ???? ???? (c) ???? v ???? (d) v ???? ???? Ans: (c) Distance of the plane from ( 2 , 1 , - 1 ) = | 2 - 2 ( 1 ) + 4 ( - 1 ) - 9 v 1 + 4 + 16 | = 13 v 21 . Q13. A unit vector perpendicular to plane determined by the points ?? ( ?? , - ?? , ?? ) , ?? ( ?? , ?? , - ?? ) and ?? ( ?? , ?? , ?? ) is (a) ???? - ?? + ?? v ?? (b) ???? + ?? + ?? v ?? (c) - ???? + ?? + ?? v ?? (d) ???? + ?? - ?? v ?? Ans: (b) We know that, ???? ? ? ? ? ? ? × ???? ? ? ? ? ? ? | ???? ? ? ? ? ? ? × ???? ? ? ? ? ? ? | ???? ? ? ? ? ? ? = ?? + ?? - 3 ?? , ???? ? ? ? ? ? ? = - ?? + 3 ?? - ?? ???? ? ? ? ? ? ? × ???? ? ? ? ? ? ? = | ?? ?? ?? 1 1 - 3 - 1 3 - 1 | = 8 ?? + 4 ?? + 4 ?? and | ???? ? ? ? ? ? ? × ???? ? ? ? ? ? ? | = 4 v 6 Hence, the unit vector is 4 ( 2 ?? + ?? + ?? ) 4 v 6 i.e. 2 ?? + ?? + ?? v 6 . Q14. A non-zero vector ?? is parallel to the line of intersection of the plane determined by the vectors ?? , ?? + ?? and the plane determined by the vectors ?? - ?? , ?? + ?? . The angle between ?? and the vector ?? - ???? + ???? is (a) ?? ?? or ?? ?? ?? (b) ?? ?? ?? or ?? ?? ?? (c) ?? ?? or ?? ?? ?? (d) None of these Ans: (a) Equation of plane containing ?? and ?? + ?? is [ ?? - ?? , ?? , ?? + ?? ] = 0 ? ( ?? - ?? ) · [ ?? × ( ?? + ?? ) ] = 0 ? [ ( ?? - 1 ) ?? + ?? ?? + ?? ?? ] · ?? = 0 ? ?? = 0 Equation of plane containing ?? - ?? and ?? + ?? is ? [ ?? - ( ?? - ?? ) ?? - ?? ? ?? + ?? ] = 0 ? ( ?? - ?? + ?? ) [ ( ?? - ?? ) × ( ?? + ?? ) ] = 0 ? ?? + ?? - ?? = 0 Let ?? = ?? 1 ?? + ?? 2 ?? + ?? 3 ?? . Since ?? is parallel to (i) and (ii) ?? 3 = 0 , ? ?? 1 + ?? 2 - ?? 3 = 0 ? ?? 1 = - ?? 2 , ?? 3 = 0 Thus a vector in the direction of ?? is ?? = ?? - ?? . If ?? is the angle between ?? and ?? - 2 ?? + 2 ?? . Then c os ? ?? = ± 1 ( 1 ) + ( - 1 ) ( - 2 ) v 1 + 1 v 1 + 4 + 4 = ± 3 v 2 · 3 ? c os ? ?? = ± 1 v 2 ? ?? = ?? / 4 or 3 ?? / 4 Q15. The sine of angle between the straight line ?? - ?? ?? = ?? - ?? ?? = ?? - ?? ?? and the plane ?? ?? - ?? ?? + ?? = ?? is (a) ?? v ?? ?? (b) v ?? ???? (c) ?? ?? v ?? (d) ???? ?? v ?? Ans: (b) We know that sin ? ?? = ???? + ???? + ???? v ?? 2 + ?? 2 + ?? 2 v ?? 2 + ?? 2 + ?? 2Read More
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