3-DIMENSIONAL GEOMETRY. 1. Find the distance between the planes, 2x + 3y +4z =4 and 4x +6y +8z =12

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1 3-DIMENSIONAL GEOMETRY 1 Find the distance between the planes, 2x + 3y +4z 4 and 4x +6y +8z 12 Ans Find the length of the perpendicular drawn from the origin to the plane 2x- 3y+6z+21 0 Ans 3 7 unit 3 Find the shortest distance of the plane 3x-4y+12z 3 from the origin Ans 3 13 unit 4 Write the intercept cut-off by plane 2x+y-z5on x-axis Ans5/2 5 Write the distance of following plane from origin, 2x-y+2z +10 Ans 1 3 unit 6 Find the value ofk, so taht the line x 2 plane 3x y-2z 7 Ans -2 y 1 6 k z+5 4 is perpendicular to the 7Write down the distance of a point P(a, b, c)fromx-axis Ans b 2 + c 2

2 8 If the Cartesian equation of a line is 3 x equation for the line y+4 2z , then write the vector Ansr (3i 4j + 3k) + ( 5i + 7j + 3k) 9 Write the equation of the straight line through the point (α, β,γ) and parallel to Z-axis Ans (αi+βj+ γk) + k 10 Find the direction cosines of the line 4 x 2 y 6 1 z 3 Ans (-2/7, 6/7, -3/7) 11 If a unit vector â makes angles π with î, π 3 4 then find the value of θ Ans π/3 with ĵ and an acute angle θ with k, 12 Find the Cartesian equation of the line which passes through the point( 2,4, 5) and is parallel to the line x+3 4 y z Ans x+2 3 y 4 5 z What are the direction cosines of a line which makes equal angles with the coordinate axes? Ans(± 1 3, ± 1 3, ± 1 3 ) 14 Write the vector equation of the line given by x 5 3 y+4 7 z 6 2 Ansr (5i 4j + 6k) + (3i + 7j + 2k) 15 Equation of a line is 4 x to the above line Ans (-2/3, 2/3, 1/3) y+3 z Find the direction cosines of a line parallel

3 16 If the equations of line AB is 3 x the line parallel to above line AB Ans (-1,-2,4) y+2 z Find the distance of a point(2, 3, 4) from X-axis Ans 5 18 Write the equations of a line parallel to the line x 2 through point (1, 2, 3) Ans x 1 z 3 y , then write the direction ratios of y+3 z Write the direction cosines of a line parallel to the line 3 x Ans (-3/7, -2/7, 6/7) 20 The equation of a line is 2x 5 parallel to this line Ans (2/7, 3/7, -6/7) 4 y+4 6 z 3 6 and passing y+2 z Find the direction cosines of the line 4 Marks Questions 21 A plane makes intercepts -6, 3, 4 respectively on the coordinate axes Find the length of the perpendicular from the origin on it Ans units 22Find the distance of the point( 1, 5, 10)from the point of intersection of the line r2i - i +2k +λ(i + 4j + 2k )and the plane r (i j + 2k ) 5 Ans 13 units 23Show that the lines 5 x y 7 z andx 8 2y z 5 are coplanar 3

4 24Find the image of the point having position vectori + 3j + 4k in the plane r (2i j + k ) Ans(-3,5,2) 25Find the vector equation of the plane through the points(2,1, 1)and( 1,3,4) and perpendicular to the plane x 2y+4z10 Ansr (18i+17j+4k) 49 26Find the coordinates of the point,where the line x 2 y+1 z intersects the planex y + z 5 0Also,find the angle between the line and the plane Ans (2, -1, 2 ), sin Find the vector equation of the plane which contains the line of intersection of the planesr(i + 2j + 3k ) 4 0 and r (2i + j k ) and which is perpendicular to the plane r (5i + 3j 6k ) Ans33x+45y+50z Find the equation of plane(s) passing through the intersection of the planes x + 3y and 3x y 4z 0 and whose perpendicular distance from origin is unity Ans 2x+y-2z+30 and x-2y-2z Find the equation of the plane passing through the point A(1,2,1) and perpendicular to the line joining points P(1,4,2) and Q(2,3,5)Also,find the distance of this plane from the line x+3 y 5 z Ans 11 units

5 30 Find the cartesian equation of the plane passing through the origin and (3, 1,2) and is parallel to the line x 4 y+3 z Ans x-19y-11z0 31Find the equation of plane that contains the point(1, 1,2) and is perpendicular to each of planes2x + 3y 2z 5 andx + 2y 3z 8 Ans 5x-4y-z Fnd the coordinates of point, where the line x+1 planex + y + 4z 6 Ans(1,1,1) y+2 z meets the 33Show that the lines r (î + ĵ k ) + λ(3î j ) andr (4î k ) + μ (2î + 3k ) intersect Also, find their point of intersection Ans (4, 0, -1) 34Find the direction cosines of a line x+2 Also, find the vector 2y 7 5 z equation of the line through the point A( 1, 2, 3) and parallel to the given line Ans (2/7, 3/7, -6/7), x+1 2 y 4 z Find the angle between the linesr 2î 5ĵ + k + λ(3î + 2ĵ + 6k ) and r 7i 6ĵ 6k + μ(î + 2ĵ + 2k ) Anscos Show that the lines x+1 their point of intersection Ans (1/2, -1/2, -3/2) y+3 z andx 2 1 y 4 3 z 6 5 intersect Also, find

6 37 Find the distance of a point (2,4-1) from the line x+5 1 y+3 z Ans 7 units 38 Show that the lines r (î + ĵ k ) + λ(3î j ) and r (4î k ) + μ (2î + 3k ) intersect Also, find their point of intersection Ans (4, 0, -1) 39Find the direction cosines of a line x+2 Also, find the vector 2y 7 5 z equation of the line through the point A( 1, 2, 3) and parallel to the given line Ans (2/7, 3/7, -6/7) and x+1 2 y 4 z Find the angle between the linesr 2î 5ĵ + k + λ(3î + 2ĵ + 6k ) and r 7i 6ĵ 6k + μ(î + 2ĵ + 2k ) 1 19 Anscos Show that the lines x+1 their point of intersection Ans (1/2, -1/2, -3/2) y+3 z andx 2 1 y 4 3 z 6 5 intersect Also, find 42 Find the value of p, so that the linesl 1 : 1 x y 5 6 z 1 5 7y 14 3 p z 3 2 and l 2 : 7 7x 3p are perpendicular to each other Also, find theequation of a line passing through a point(3,2, -4) and parallel to line l 1 Ans 3 x 3 y 2 z A line passes through the point (2, -1, 3) and is perpendicular to the linesr (i + j - k ) + x (2i - 2j + k )equation in vector and Cartesian forms Ans 2 x 6 y 1 z 3 3 6

7 44 Find the shortest distance between the lineswhosevector equations are r i + j +x(2i -j + k )and r 2i + j - k + µ(3i - 5j + 2k) Ans 10/ 59 units 45 Find the shortest distance between the twolines whose vector equations are r (i + 2j + 3k ) + x(i - 3j + 2k ) andr (4i + 5j + k ) 9 Ans unit Find the shortest distance between the following lines: x+1 y 5 z Ans y+1 z+1, x Find the distance between the lines l 1 and l 2 given by l 1 : r i + 2j 4k + x(2i + 3j + 6k ),l 2 :r 3i + 3j 5k + µ(4i + 6j + 12k ) Ans 48 Find the vector and Cartesian equations of the line passing through the point (2, 1, 3) and perpendicular to the lines x 1 y 2 z 3 and x y z Ansr 2i + j + 3k + (2i + 7j + 6k) 49 The Cartesian equation of a line is 6x 2 3y + 1 2z -2 Find the direction cosines of the line Write down the Cartesian and vector equations of a line passing through (2, -1, -1) which are parallel to the given line Ansr (2i j k) + ( 1 6 i j k) 50 Find the shortest distance between the two lines whose vector equations are r (6i + 2j ) + x(i 2j + 2k ) andr (-4i k) + µ(3i 2j + 2k ) Ans 9 units 51 Show that the lines r 3i + 2j 4k + x(i + 2j + 2k ); r 5i 2j + µ(3i + 2j + 6k )are intersecting Hence, find their point of intersection

8 Ans (-1, -6, -12) 52 Find the vector and Cartesian equations of line passing through point (1, 2, -4) and perpendicular to two lines x 8 y+19 z Ans r (i+2j-4k)+ (2i+3j+6k) and x 15 3 y 29 8 z Find the angle between following pair of lines x+2 z 5 4 Ans π/2 and check whether the lines are parallel or perpendicular 2 y 1 z+3 x+2 and 2y Find the shortest distance between lines whose vector equations are r (1 t)i + (t 2)j + (3 2t)k and r (s + 1)i + (2s 1)j - (2s + 1)k Ans 8 29 units Find shortest distance between the lines r (i + 2j + k ) + x(i j + k ) and r (2i j k ) + µ(2i + j + 2k ) Ans units 56Find the equation of the perpendicular from point (3, 1,11) to line x 2 y 2 z 3 Also find the coordinates of foot of perpendicular and the length of 4 perpendicular Ans 53 units 57Find the perpendicular distance of point (1,0,0) from the lines x 1 z+10 8 Also, find the coordinates of foot of perpendicular and equation of perpendicular Ans 53 units,(3,4, -2) and x 1 2 y 4 z 2 y

9 58 Find the points on the line x+2 point P(1,3,3) Ans (-2, -1, 3) and (4,3,7) y+1 z Find the shortest distance between the lines l 1 : x 1 z 1 2 Ans 3/ 2 units at a distance of 5 units from the y z 1 1 l 2: x 2 2 y Find the shortest distance between lines r (1 + 2λ)i + (1 λ)j + λk 2i j k ) + μ(2i + j + 2k ) r Ans 3/ 29units 61 Find the value of λ, so that following lines are perpendicular to each other x+5 2 y 1 z and x 2y+1 1 z 5λ λ 3 Ans 1 62 Find the value of λ so that lines 1 x perpendicular to each other Ans -2 63Find the value of λ so that lines 1 x perpendicular to each other Ans 7 y 2 3 2λ 7y λ z 3 2 5z and x+1 3λ and 7 7x 3λ y z 7 are y z 5 are 64Find the length and foot of perpendicular drawn from the point (2, 1,5) to line x 11 y+2 z Ans 14units, (1,2,3)

10 6MARKS QUESTIONS 65Find the equation of the plane passing through the line of intersection of the plane r (i + j + k ) 1 and r (2i + 3j k) and parallel to X-axis Ansr (i -3k) Find the distance between the point(7,2,4)and the plane determined by the points A(2,5, 3), B( 2, 3,5) and C(5,3, 3) Ans 29 units 67Find the equation of the plane through the line of intersection of the planesx + y + z 1and2x + 3y + 4z 5,which is perpendicular to the planex y + z 0Also,find the distance of the plane obtained above, from the origin Ans x z+ 20, distance 2 units 68Find the distance of the point(2,12,5)from the point of intersection of the liner 2i 4j + 2k + λ(3i + 4j + 2k ) and the planer (i 2j + k ) 0 Ans 13 units 69Find the equation of the plane that contains the point(1, 1,2) and is perpendicular to both the planes 2x + 3y 2z 5 and x + 2y 3z 8Hence, find the distance of point P( 2,5,5)from the plane obtained above Ans 42 units 70Show that the linesr (i + j k ) + λ(3i j ) and r (4i k ) + μ(2i + 3k ) are coplanar Also, find the equation of the plane containing them Ans 3x+9y-2z14

11 71Find the distance of the point(1, 2,3)from the plane x y + z 5 measured parallel to the line x 1 y 3 z Ans units 72Find the equation of the plane passing through the line of intersection of the planes r (i + 3j ) 6 0andr (3i j 4k 0,whose perpendicular distance from origin is unity Ansr (-2i+4j+4k) Find the vector equation of the line passing through the point(1,2,3) and parallel to the planesr (i j + 2k ) 5 and r (3i + j + k ) 6 Ansr (i + 2j + 3k) + ( 3i + 5j + 4k) 74Find the coordinates of the point, where the line through(3 4, 5) and (2, 3,1) crosses the plane, passing the points (2,2,1), (3,0,1) and (4, 1,0) Ans (1,-2,7) 75 Find the vector equation of the plane passing through the three points with position vectorsi + j 2k, 2i j + k and i + 2j + k Also,find the coordinates of the point of intersection of this plane and the line r 3i j k + λ(2i 2j + k ) Ansr i + j 2k 76Find the equation of the plane determined by pointsa(3, 1,2), B(5,2,4) and C( 1, 1,6)and hence find the distance between plane and point(6,5,9)

12 Ans 6 34 units 77Find the length and foot of perpendicular from the pointp(7,14,5)to plane 2x + 4y z 2Also,find the image of the point P in the plane Ans 189 unitsand(-5,-10,11) 78Find the equation of the plane which contains the line of intersection of planes r (i + 2j + 3k ) 4 0, r (2i + j k ) and which is perpendicular to plane r (5i + 3j 6k )8 0 Ansr, (33i + 45j + 50k) 41 79Find the equation of the plane passing through the line of intersection of planes 2x + y z 3and5x 3y + 4z and parallel to line x 1 y 3 z Ans 7x+9y-10z Find the equation of plane passing through the point( 1,3,2) and perpendicular to each of the planesx + 2y + 3z 5 and 3x + 3y + z 5 Ans7x-8y+3z Find the vector equation of plane passing through the pointsa(2,2, 1), B(3,4,2) and C(7,0,6)Also,find the Cartesian equation of plane Ansr (5i + 2j 3k) 17 82Find the equation of the plane passing through point(1,1, 1) and perpendicular to planes x + 2y + 3z 7 0 and 2x 3y 4z 0 Ans7x+2y-7z-260

13 83 Find the vector and cartesian equation of a plane containing the two linesr (2i + j 3k ) + λ (i + 2j + 5k ) and r (3i + 3j + 2k ) + μ(3i 2j + 5k ) Also,show that r (2i + 5j + 2k ) + P(3i 2j + 5k ) lies in the plane Ansr (10i + 5j 4k) 37 and 10x + 5y 4z 37 84Find the equation of the plane passing through the point(1,2,1) and perpendicular to line joining points(1,4,2) and (2,3,5)Also,find the coordinates of foot of the perpendicular and the perpendicular distance of the point(4,0,3) from above found plane Ans x-y+3z-20, (3,1,0), Distance 11 units 85Find the equation of the plane passing through point P(1,1,1) and containing the liner ( 3i + j + 5k ) + λ(3i j 5k )Also,show that plane contains the liner + ( i + 2j + 5k ) + μ(i 2j 5k ) Ansr (i 2j + k) 0 86Find the coordinates of the foot of the perpendicular, distance of pointp(3,2,1) from the plane 2x y + z + 1 0Also,find image of the point in the plane Ans(1,3,0) Distance 6 units Image Point (-1,4,1) 87Find the distance of the point(2,3,4)from the line x+3 3 y 2 parallel to the parallel to the plane3x + 2y + 2z 5 0 Ans 33 units 6 z 2 measured

14 88Find the distance of point( 2,3, 4)from the line x+2 parallel to the plane4x + 12y 3z Ans 17/2 units 2y z+4 5 measured 89Find the coordinates of image of point (1, 3, 4) in the plane 2x y + z Ans(-3,5,2) 90 From the point (1, 2, 4), a perpendicular is drawn on the plane 2x + y 2z Find the equation, the length and the coordinates of foot of perpendicular Ans x 1 2 y 2 1 z 4 2, 1/3 unit and(11/9, 19/9, 34/9) 91Find the equation of plane passing through the point ( 1, 1, 2) and perpendicular to each plane 2x + 3y 3z 2and 5x 4y + z 6 Ans 9x+17y+23z Find the equations of plane passing through points (3, 4, 1) and (0, 1, 0) and parallel to line x+3 y 3 z Ans 8x-13y+15z Find the distance of the point P( 1, 5, 10) from the point of intersection of the line joining the points A(2, 1,2) and B(5,3,4) with the plane x y + z 5 Ans 13 units 94 Find the vector and Cartesian forms of the equation of the plane passing through the point (1,2, 4)and parallel to the lines r i + 2j 4k + λ(2i + 3j + 6k )and r i 3j + 5k + μ(i + j k )Also, fnd the distance of the point (9, 8, 10) from the plane thus obtained

15 Ans 146units 95 Find the equation of line passing through points A(0,6, 9) and B( 3, 6,3) If D is the foot of perpendicular drawn from the point C(7,4, 1) on the line AB, then find the coordinates of point D and equation of line CD Ansline x 7 y 4 z Find the image of the point (1,6,3) on the line x 1 y 1 z Also, Write the equation of the line joining the given points and its image and find The length of segment joining given point and its image Ans 52 units 97 Write the vector equations of following lines and hence find the distance between them x 1 y 2 z+4, x 3 y 3 z Ans The points A(4,5,10), B(2,3,4) and C(1,2, 1) are the three vertices of parallelogram ABCD Find the vector equations of sides AB and BC and also find the coordinates of point D Ansr (4i+5j+10k) - (2i+2j+6k) and r (2i+3j+4k) -µ(i+j+5k) 99 Find the coordinates of foot of perpendicular drawn from the point (0,2,3) on the line x+3 y 1 z+4 Also, find the length of perpendicular Ans 21 units 100Find the perpendicular distance of the point (2,3,4) from the line 4 x y z Also, find coordinates of foot of perpendicular 3 Ans (170/49, 78/49, 60/49) 101: If the equations of a line AB are 3 x y+2 z+2, find the direction cosines of a line parallel to AB 3 2 6

16 Ans: 3, 2, : The Cartesian equations of a line are 6x-23y+12z-2 Find the direction cosines of the line Write down the Cartesian and vector equations of the line passing through (2,-1,-1) which is parallel to the given line Ans 1 14, 2 14, : Show that the lines r i +j -k + λ (3i -j ) and r 4i -k + µ(2i +k ) intersect Also find their point of intersection Ans: (4, 0, -1) 104: Find the shortest distance between the lines through the points A (6, 2, 2) and A (-4, 0, -1) in the direction 1, -2, 2 and 3, -2, -2 respectively Ans: 9 units 105: Find the shortest distance between the lines x 1 y 2 z and x 2 3 y 3 4 z 5 5 Ans: 6 6 units 106: By computing the shortest distance between the following pair of lines, determine whether they intersect or not? r i - j +λ (2i -k ), r 2i -j +µ(i -j -k ) Ans: units, No 107: Show that the lines x a+d α j y a α z a d α+j and x b+c β γ γ b β z b c β+γ are coplanar

17 108: A line with direction ratios 2, 7, -5 is drawn to intersect the lines x 5 y 7 z+2 x+3 and y 3 z Find the co-ordinates of the points of intersection and the length intercepted on it Ans: (2, 8, -3), (0, 1, 2), 78 units 109: Show that the line x 1 y z 3 x 4 and y 1 z intersect Find their point of intersection Ans: (-1, -1, -1) 110: Find the image of the point (1, 2, 3) in the line x 6 3 y 7 2 z 7 2 Ans: (5, 8, 15)

THREE DIMENSIONAL GEOMETRY

For more important questions visit : www4onocom CHAPTER 11 THREE DIMENSIONAL GEOMETRY POINTS TO REMEMBER Distance between points P(x 1 ) and Q(x, y, z ) is PQ x x y y z z 1 1 1 (i) The coordinates of point