JEE Main & Advanced Mathematics Three Dimensional Geometry Question Bank

done Line Total Questions - 43

• question_answer1)
  Distance of the point \[({{x}_{1}},{{y}_{1}},{{z}_{1}})\] from the line\[\frac{x-{{x}_{2}}}{l}=\frac{y-{{y}_{2}}}{m}=\frac{z-{{z}_{2}}}{n}\], where \[l,\]m and n are the direction cosines of line is
  A)
                      \[\sqrt{{{({{x}_{1}}-{{x}_{2}})}^{2}}+{{({{y}_{1}}-{{y}_{2}})}^{2}}+{{({{z}_{1}}-{{z}_{2}})}^{2}}-{{[l({{x}_{1}}-{{x}_{2}})+m({{y}_{1}}-{{y}_{2}})+n({{z}_{1}}-{{z}_{2}})]}^{2}}}\] done clear
  B)
                                                                \[\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}+{{({{z}_{2}}-{{z}_{1}})}^{2}}}\] done clear
  C)
                                                                \[\sqrt{({{x}_{2}}-{{x}_{1}})l+({{y}_{2}}-{{y}_{1}})m+({{z}_{2}}-{{z}_{1}})n}\] done clear
  D)
                                                                None of these done clear
  View Solution play_arrow

• question_answer2)
  If the co-ordinates of the points A, B, C, D be (1, 2, 3),         (4, 5, 7), (?4, 3, ?6) and (2, 9, 2) respectively, then the angle between the lines AB and CD is
  A)
                                                                \[\frac{\pi }{6}\] done clear
  B)
                                                                \[\frac{\pi }{4}\] done clear
  C)
                                                                \[\frac{\pi }{3}\] done clear
  D)
                                                                None of these done clear
  View Solution play_arrow

• question_answer3)
  The angle between the lines \[\frac{x}{1}=\frac{y}{0}=\frac{z}{-1}\] and \[\frac{x}{3}=\frac{y}{4}=\frac{z}{5}\] is [Pb. CET 2002]
  A)
                                                                \[{{\cos }^{-1}}\frac{1}{5}\] done clear
  B)
                                                                \[{{\cos }^{-1}}\frac{1}{3}\] done clear
  C)
                                                                \[{{\cos }^{-1}}\frac{1}{2}\] done clear
  D)
                                                                \[{{\cos }^{-1}}\frac{1}{4}\] done clear
  View Solution play_arrow

• question_answer4)
  If \[\frac{x-1}{l}=\frac{y-2}{m}=\frac{z+1}{n}\]is the equation of the line through   (1, 2, ?1) and (?1, 0, 1), then (l, m, n) is          [MP PET 1992]
  A)
                                                                (?1, 0, 1) done clear
  B)
                                                                (1, 1, ?1) done clear
  C)
                                                                (1, 2, ?1) done clear
  D)
                                                                (0, 1, 0) done clear
  View Solution play_arrow

• question_answer5)
  If the angle between the lines whose direction ratios are         2,?1 , 2 and a,  3, 5 be \[45{}^\circ \], then a =
  A)
                                                                1 done clear
  B)
                                                                2 done clear
  C)
                                                                3 done clear
  D)
                                                                4 done clear
  View Solution play_arrow

• question_answer6)
  The point of intersection of lines \[\frac{x-4}{5}=\] \[\frac{y-1}{2}=\frac{z}{1}\] and \[\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\] is [AISSE 1986; AMU 2005]
  A)
                                                                (?1, ?1, ?1) done clear
  B)
                                                                (?1, ?1, 1) done clear
  C)
                                                                (1, ?1, ?1) done clear
  D)
                                                                (?1, 1, ?1) done clear
  View Solution play_arrow

• question_answer7)
  Direction ratios of two lines are a, b, c and \[\frac{1}{bc},\frac{1}{ca},\frac{1}{ab}\]. The lines are
  A)
                                                                Mutually perpendicular done clear
  B)
                                                                Parallel done clear
  C)
                                                                Coincident done clear
  D)
                                                                None of these done clear
  View Solution play_arrow

• question_answer8)
  The angle between the lines whose direction cosines are proportional to (1, 2, 1) and (2, ?3, 6) is
  A)
                                                                \[{{\cos }^{-1}}\left( \frac{2}{7\sqrt{6}} \right)\] done clear
  B)
                                                                \[{{\cos }^{-1}}\left( \frac{1}{7\sqrt{6}} \right)\] done clear
  C)
                                                                \[{{\cos }^{-1}}\left( \frac{3}{7\sqrt{6}} \right)\] done clear
  D)
                                                                \[{{\cos }^{-1}}\left( \frac{5}{7\sqrt{6}} \right)\] done clear
  View Solution play_arrow

• question_answer9)
  Direction ratios of the line represented by the equation \[x=ay+b,\] \[z=cy+d\] are
  A)
                                                                (a, 1, c) done clear
  B)
                                                                (a, b ? d, c) done clear
  C)
                                                                (c, 1, a) done clear
  D)
                                                                (b, ac, d) done clear
  View Solution play_arrow

• question_answer10)
  The equation of a line passing through the point  (?3, 2, ? 4) and equally inclined to the axes, are
  A)
                                                                \[x-3=y+2=z-4\] done clear
  B)
                                                                \[x+3=y-2=z+4\] done clear
  C)
                                                                \[\frac{x+3}{1}=\frac{y-2}{2}=\frac{z+4}{3}\] done clear
  D)
                                                                None of these done clear
  View Solution play_arrow

• question_answer11)
  The co-ordinates of the foot of perpendicular drawn from the origin to the line joining the points (?9, 4, 5) and (10, 0, ?1) will be
  A)
                                                                (? 3, 2, 1) done clear
  B)
                                                                (1, 2, 2) done clear
  C)
                                                                (4, 5, 3) done clear
  D)
                                                                None of these done clear
  View Solution play_arrow

• question_answer12)
  The symmetric equation of lines \[3x+2y+z-5=0\] and \[x+y-2z-3=0\], is
  A)
                                                                \[\frac{x-1}{5}=\frac{y-4}{7}=\frac{z-0}{1}\] done clear
  B)
                                                                \[\frac{x+1}{5}=\frac{y+4}{7}=\frac{z-0}{1}\] done clear
  C)
                                                                \[\frac{x+1}{-5}=\frac{y-4}{7}=\frac{z-0}{1}\] done clear
  D)
                                                                \[\frac{x-1}{-5}=\frac{y-4}{7}=\frac{z-0}{1}\] done clear
  View Solution play_arrow

• question_answer13)
  The angle between the lines whose direction cosines satisfy the equations \[l+m+n=0\], \[{{l}^{2}}+{{m}^{2}}-{{n}^{2}}=0\] is given by [MP PET 1993; RPET 2001]
  A)
                                                                \[\frac{2\pi }{3}\] done clear
  B)
                                                                \[\frac{\pi }{6}\] done clear
  C)
                                                                \[\frac{5\pi }{6}\] done clear
  D)
                                                                \[\frac{\pi }{3}\] done clear
  View Solution play_arrow

• question_answer14)
  The equation of straight line passing through the points          (a, b, c) and (a ? b, b? c, c ? a), is [MP PET 1994]
  A)
                                                                \[\frac{x-a}{a-b}=\frac{y-b}{b-c}=\frac{z-c}{c-a}\] done clear
  B)
                                                                \[\frac{x-a}{b}=\frac{y-b}{c}=\frac{z-c}{a}\] done clear
  C)
                                                                \[\frac{x-a}{a}=\frac{y-b}{b}=\frac{z-c}{c}\] done clear
  D)
                                                                \[\frac{x-a}{2a-b}=\frac{y-b}{2b-c}=\frac{z-c}{2c-a}\] done clear
  View Solution play_arrow

• question_answer15)
  The equation of straight line passing through the point (a, b, c) and parallel to z- axis, is         [MP PET 1995; Pb. CET 2000]
  A)
                                                                \[\frac{x-a}{1}=\frac{y-b}{1}=\frac{z-c}{0}\] done clear
  B)
                                                                \[\frac{x-a}{0}=\frac{y-b}{1}=\frac{z-c}{1}\] done clear
  C)
                                                                \[\frac{x-a}{1}=\frac{y-b}{0}=\frac{z-c}{0}\] done clear
  D)
                                                                \[\frac{x-a}{0}=\frac{y-b}{0}=\frac{z-c}{1}\] done clear
  View Solution play_arrow

• question_answer16)
  The length of the perpendicular drawn from the point            (5, 4, ?1) on the line \[\frac{x-1}{2}=\frac{y}{9}=\frac{z}{5}\] is
  A)
                                                                \[\sqrt{\frac{110}{2109}}\] done clear
  B)
                                                                \[\sqrt{\frac{2109}{110}}\] done clear
  C)
                                                                \[\frac{2109}{110}\] done clear
  D)
                                                                54 done clear
  View Solution play_arrow

• question_answer17)
  The length of the perpendicular from point (1, 2, 3) to the line \[\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\]is [MP PET 1997]
  A)
                                                                5 done clear
  B)
                                                                6 done clear
  C)
                                                                7 done clear
  D)
                                                                8 done clear
  View Solution play_arrow

• question_answer18)
  The angle between the lines whose direction cosines are connected by the relations \[l+m+n=0\] and \[2lm+2nl-mn=0\],  is
  A)
                                                                \[\frac{\pi }{3}\] done clear
  B)
                                                                \[\frac{2\pi }{3}\] done clear
  C)
                                                                \[\pi \] done clear
  D)
                                                                None of these done clear
  View Solution play_arrow

• question_answer19)
  The perpendicular distance of the point (2, 4, ?1) from the line \[\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}\] is [Kurukshetra CEE 1996]
  A)
                                                                3 done clear
  B)
                                                                5 done clear
  C)
                                                                7 done clear
  D)
                                                                9 done clear
  View Solution play_arrow

• question_answer20)
  The angle between two lines \[\frac{x+1}{2}=\]\[\frac{y+3}{2}=\frac{z-4}{-1}\] and \[\frac{x-4}{1}=\frac{y+4}{2}=\frac{z+1}{2}\] is [MP PET 1996]
  A)
                                                                \[{{\cos }^{-1}}\left( \frac{1}{9} \right)\] done clear
  B)
                                                                \[{{\cos }^{-1}}\left( \frac{2}{9} \right)\] done clear
  C)
                                                                \[{{\cos }^{-1}}\left( \frac{3}{9} \right)\] done clear
  D)
                                                                \[{{\cos }^{-1}}\left( \frac{4}{9} \right)\] done clear
  View Solution play_arrow

• question_answer21)
  The straight lines \[\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}\]                                                                                     and \[\frac{x-1}{2}=\frac{y-2}{2}=\frac{z-3}{-2}\] are
  A)
                                                                Parallel lines done clear
  B)
                                                                Intersecting at \[60{}^\circ \] done clear
  C)
                                                                Skew lines done clear
  D)
                                                                Intersecting at right angle done clear
  View Solution play_arrow

• question_answer22)
  The equation of the line passing through the points ( 3, 2, 4) and (4, 5, 2) is
  A)
                                                                \[\frac{x+3}{1}=\frac{y+2}{3}=\frac{z+4}{-2}\] done clear
  B)
                                                                \[\frac{x-3}{1}=\frac{y-2}{3}=\frac{z-4}{-2}\] done clear
  C)
                                                                \[\frac{x+3}{7}=\frac{y+2}{7}=\frac{z+4}{6}\] done clear
  D)
                                                                \[\frac{x-3}{7}=\frac{y-2}{7}=\frac{z-4}{6}\] done clear
  View Solution play_arrow

• question_answer23)
  The angle between the lines \[\frac{x+4}{1}=\frac{y-3}{2}=\frac{z+2}{3}\] and \[\frac{x}{3}=\frac{y-1}{-2}=\frac{z}{1}\] is
  A)
                                                                \[{{\sin }^{-1}}\left( \frac{1}{7} \right)\] done clear
  B)
                                                                \[{{\cos }^{-1}}\left( \frac{2}{7} \right)\] done clear
  C)
                                                                \[{{\cos }^{-1}}\left( \frac{1}{7} \right)\] done clear
  D)
                                                                None of these done clear
  View Solution play_arrow

• question_answer24)
  The angle between the pair of lines with direction ratios       (1, 1, 2) and \[(\sqrt{3}-1,-\sqrt{3}-1,4)\] is       [MP PET 1997, 2000]
  A)
                                                                \[30{}^\circ \] done clear
  B)
                                                                \[45{}^\circ \] done clear
  C)
                                                                \[60{}^\circ \] done clear
  D)
                                                                \[90{}^\circ \] done clear
  View Solution play_arrow

• question_answer25)
  The acute angle between the line joining the points (2,1,?3), (?3,1,7) and a line parallel to \[\frac{x-1}{3}=\] \[\frac{y}{4}=\frac{z+3}{5}\] through the point (?1, 0, 4) is [MP PET 1998]
  A)
                                                                \[{{\cos }^{-1}}\left( \frac{7}{5\sqrt{10}} \right)\] done clear
  B)
                                                                \[{{\cos }^{-1}}\left( \frac{1}{\sqrt{10}} \right)\] done clear
  C)
                                                                \[{{\cos }^{-1}}\left( \frac{3}{5\sqrt{10}} \right)\] done clear
  D)
                                                                \[{{\cos }^{-1}}\left( \frac{1}{5\sqrt{10}} \right)\] done clear
  View Solution play_arrow

• question_answer26)
  The angle between the straight lines \[\frac{x+1}{2}=\frac{y-2}{5}=\frac{z+3}{4}\] and  \[\frac{x-1}{1}=\frac{y+2}{2}=\frac{z-3}{-3}\] is [MP PET 2000]
  A)
                                                                \[45{}^\circ \] done clear
  B)
                                                                \[30{}^\circ \] done clear
  C)
                                                                \[60{}^\circ \] done clear
  D)
                                                                \[90{}^\circ \] done clear
  View Solution play_arrow

• question_answer27)
  If direction ratios of two lines are \[5,\,\,-12,\,13\] and \[-3,\,4,\,5\] then the angle between them is [RPET 2001]
  A)
                                                                \[{{\cos }^{-1}}(1/65)\] done clear
  B)
                                                                \[{{\cos }^{-1}}(2/65)\] done clear
  C)
                                                                \[{{\cos }^{-1}}(3/65)\] done clear
  D)
                                                                \[\pi /2\] done clear
  View Solution play_arrow

• question_answer28)
  The shortest distance between the lines \[\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}\] and \[\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}\] is [RPET 2001; MP PET 2002]
  A)
                                                                \[\sqrt{30}\] done clear
  B)
                                                                \[2\sqrt{30}\] done clear
  C)
                                                                \[5\sqrt{30}\] done clear
  D)
                                                                \[3\sqrt{30}\] done clear
  View Solution play_arrow

• question_answer29)
  If the direction ratios of two lines are given by \[3lm-4\,ln+mn=0\] and \[l+2m+3n=0\], then the angle between the lines is [EAMCET 2003]
  A)
                                                                \[\pi /2\] done clear
  B)
                                                                \[\pi /3\] done clear
  C)
                                                                \[\pi /4\] done clear
  D)
                                                                \[\pi /6\] done clear
  View Solution play_arrow

• question_answer30)
  Equation of x-axis is [MP PET 2002]
  A)
                                                                \[\frac{x}{1}=\frac{y}{1}=\frac{z}{1}\] done clear
  B)
                                                                \[\frac{x}{0}=\frac{y}{1}=\frac{z}{1}\] done clear
  C)
                                                                \[\frac{x}{1}=\frac{y}{0}=\frac{z}{0}\] done clear
  D)
                                                                \[\frac{x}{0}=\frac{y}{0}=\frac{z}{1}\] done clear
  View Solution play_arrow

• question_answer31)
  The straight line \[\frac{x-3}{3}=\frac{y-2}{1}=\frac{z-1}{0}\]is [RPET 2002]
  A)
                                                                Parallel to x-axis done clear
  B)
                                                                Parallel to y-axis done clear
  C)
                                                                Parallel to z-axis done clear
  D)
                                                                Perpendicular to z-axis done clear
  View Solution play_arrow

• question_answer32)
  The angle between a line with direction ratios 2 : 2 : 1 and a line joining (3, 1, 4) to (7, 2, 12) is                     [DCE 2002]
  A)
                                                                \[{{\cos }^{-1}}(2/3)\] done clear
  B)
                                                                \[{{\cos }^{-1}}(-2/3)\] done clear
  C)
                                                                \[{{\tan }^{-1}}(2/3)\] done clear
  D)
                                                                None of these done clear
  View Solution play_arrow

• question_answer33)
  The line \[\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}\] and \[\frac{x-1}{k}=\] \[\frac{y-4}{2}=\frac{z-5}{1}\] are coplanar, if [AIEEE 2003]
  A)
                                                                \[k=0\]or ?1 done clear
  B)
                                                                \[k=0\]or 1 done clear
  C)
                                                                \[k=0\]or ?3 done clear
  D)
                                                                \[k=3\]or ?3 done clear
  View Solution play_arrow

• question_answer34)
  If direction cosines of two lines are proportional to (2, 3, ?6) and (3, ?4, 5), then the acute angle between them is                                                                                           [MP PET  2003]
  A)
                                                                \[{{\cos }^{-1}}\left( \frac{49}{36} \right)\] done clear
  B)
                                                                \[{{\cos }^{-1}}\left( \frac{18\sqrt{2}}{35} \right)\] done clear
  C)
                                                                \[96{}^\circ \] done clear
  D)
                                                                \[{{\cos }^{-1}}\left( \frac{18}{35} \right)\] done clear
  View Solution play_arrow

• question_answer35)
  The equation to the straight line passing through the points (4, ?5, ?2) and (?1, 5, 3) is [MP PET 2003]
  A)
                                                                \[\frac{x-4}{1}=\frac{y+5}{-2}=\frac{z+2}{-1}\] done clear
  B)
                                                                \[\frac{x+1}{1}=\frac{y-5}{2}=\frac{z-3}{-1}\] done clear
  C)
                                                                \[\frac{x}{-1}=\frac{y}{5}=\frac{z}{3}\] done clear
  D)
                                                                \[\frac{x}{4}=\frac{y}{-5}=\frac{z}{-2}\] done clear
  View Solution play_arrow

• question_answer36)
  If \[A,B,C,D\]are the points (2, 3, ?1),(3, 5, ?3), (1, 2, 3),       (3, 5, 7) respectively, then the angle between AB and CD is                                                                                    [Orissa JEE 2003]
  A)
                                                                \[\frac{\pi }{2}\] done clear
  B)
                                                                \[\frac{\pi }{3}\] done clear
  C)
                                                                \[\frac{\pi }{4}\] done clear
  D)
                                                                \[\frac{\pi }{6}\] done clear
  View Solution play_arrow

• question_answer37)
  The point of intersection of the lines \[\frac{x-5}{3}=\frac{y-7}{-1}=\frac{z+2}{1},\] \[\frac{x+3}{-36}=\frac{y-3}{2}=\frac{z-6}{4}\] is [MP PET 2004]
  A)
                                                                \[21,\,\frac{5}{3},\frac{10}{3}\] done clear
  B)
                                                                \[(\,2,\,10,\,4)\] done clear
  C)
                                                                \[(-3,\,3,\,6)\] done clear
  D)
                                                                \[(5,\,7,\,-2)\] done clear
  View Solution play_arrow

• question_answer38)
  A line makes the same angle\[\theta \], with each of the x and z?axis. If the angle \[\beta \], which it makes with y-axis is such that \[{{\sin }^{2}}\beta =3{{\sin }^{2}}\theta ,\]then \[{{\cos }^{2}}\theta \]equals  [AIEEE 2004]
  A)
                                                                \[\frac{3}{5}\] done clear
  B)
                                                                \[\frac{2}{3}\] done clear
  C)
                                                                \[\frac{1}{5}\] done clear
  D)
                                                                None of these done clear
  View Solution play_arrow

• question_answer39)
  The angle between the lines \[2x=3y=-z\] and \[6x=-y=-4z\],  is [MP PET 1994, 99; AIEEE 2005]
  A)
                                                                \[0{}^\circ \] done clear
  B)
                                                                \[30{}^\circ \] done clear
  C)
                                                                \[45{}^\circ \] done clear
  D)
                                                                \[90{}^\circ \] done clear
  View Solution play_arrow

• question_answer40)
  If the lines \[\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}\], \[\frac{x-1}{3k}=\frac{y-5}{1}=\frac{z-6}{-5}\] are at right angles, then k = [MP PET 1997, 2001]
  A)
                                                                ?10 done clear
  B)
                                                                \[\frac{10}{7}\] done clear
  C)
                                                                \[\frac{-10}{7}\] done clear
  D)
                                                                \[\frac{-7}{10}\] done clear
  View Solution play_arrow

• question_answer41)
  The direction cosines of three lines passing through the origin are \[{{l}_{1}},{{m}_{1}},{{n}_{1}};\,{{l}_{2}},{{m}_{2}},{{n}_{2}}\]and \[{{l}_{3}},{{m}_{3}},{{n}_{3}}\]. The lines will be coplanar, if
  A)
                                                                \[\left| \,\begin{matrix}    {{l}_{1}} & {{n}_{1}} & {{m}_{1}}  \\    {{l}_{2}} & {{n}_{2}} & {{m}_{2}}  \\    {{l}_{3}} & {{n}_{3}} & {{m}_{3}}  \\ \end{matrix}\, \right|=0\] done clear
  B)
                                                                \[\left| \,\begin{matrix}    {{l}_{1}} & {{m}_{2}} & {{n}_{3}}  \\    {{l}_{2}} & {{m}_{3}} & {{n}_{1}}  \\    {{l}_{3}} & {{m}_{1}} & {{n}_{2}}  \\ \end{matrix}\, \right|=0\] done clear
  C)
                                                                \[{{l}_{1}}{{l}_{2}}{{l}_{3}}+{{m}_{1}}{{m}_{2}}{{m}_{3}}+{{n}_{1}}{{n}_{2}}{{n}_{3}}=0\] done clear
  D)
                                                                None of these done clear
  View Solution play_arrow

• question_answer42)
  The distance of the point (2, 3, 4) from the line \[1-x=\frac{y}{2}=\frac{1}{3}(1+z)\] is [J & K 2005]
  A)
                                                                \[\frac{1}{7}\sqrt{35}\] done clear
  B)
                                                                \[\frac{4}{7}\sqrt{35}\] done clear
  C)
                                                                \[\frac{2}{7}\sqrt{35}\] done clear
  D)
                                                                \[\frac{3}{7}\sqrt{35}\] done clear
  View Solution play_arrow

• question_answer43)
  The angle between the straight lines \[\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}\]and \[\frac{x+1}{-1}=\frac{y-4}{8}=\frac{z-5}{4}\]is [DCE 2005]
  A)
                                                                \[{{\cos }^{-1}}\left( \frac{13}{9\sqrt{38}} \right)\] done clear
  B)
                                                                \[{{\cos }^{-1}}\left( \frac{26}{9\sqrt{38}} \right)\] done clear
  C)
                                                                \[{{\cos }^{-1}}\left( \frac{4}{\sqrt{38}} \right)\] done clear
  D)
                                                                \[{{\cos }^{-1}}\left( \frac{2\sqrt{2}}{\sqrt{19}} \right)\] done clear
  View Solution play_arrow