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Three Dimensional Geometry

Short Answer Type

81. If the lines

fraction numerator straight x minus 1 over denominator negative 3 end fraction space equals space fraction numerator straight y minus 2 over denominator 2 straight k end fraction space equals space fraction numerator straight z minus 3 over denominator 2 end fraction space and space fraction numerator straight x minus 1 over denominator 3 straight k end fraction space equals space fraction numerator straight y minus 1 over denominator 1 end fraction space equals space fraction numerator straight z minus 6 over denominator negative 5 end fraction

are prependicular, then find the value of k.

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82.

Show that the lines x = ay + b, z = cy + d and x = a' y + b' , z = c' y + d' are perpendicular to each other, if aa' + cc' + 1 = 0.

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83. If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (– 4, 3, –6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.

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Long Answer Type

84. Find the equations of the line passing through the point P(–1, 3, –2) and perpendicular to the lines

straight x over 1 space equals space straight y over straight z space equals space straight z over 3 space space and space fraction numerator straight x plus 2 over denominator negative 3 end fraction space equals space fraction numerator straight y minus 1 over denominator 2 end fraction space equals space fraction numerator straight z plus 1 over denominator 5 end fraction

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85. Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to the two lines:

fraction numerator straight x minus 8 over denominator 3 end fraction space space equals space fraction numerator straight y plus 19 over denominator negative 16 end fraction space equals space fraction numerator straight z minus 10 over denominator 7 end fraction space and space fraction numerator straight x minus 15 over denominator 3 end fraction space equals space fraction numerator straight y minus 29 over denominator 8 end fraction space equals space fraction numerator straight z minus 5 over denominator negative 5 end fraction.

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86. Show that, if the axes are rectangular, then the equations of the line through (α, β, γ) right angles to the lines
$straight x over straight l subscript 1 space equals space straight y over straight m subscript 1 space equals space straight z over straight n subscript 1 comma space space straight x over straight l subscript 2 space equals space straight y over straight m subscript 2 space equals space straight z over straight n subscript 2$

are $fraction numerator straight x minus straight alpha over denominator straight m subscript 1 space straight n subscript 2 minus straight m subscript 2 straight n subscript 1 end fraction space equals space fraction numerator straight y minus straight beta over denominator straight n subscript 1 straight l subscript 2 minus straight n subscript 2 straight l subscript 1 end fraction space equals fraction numerator straight z minus straight gamma over denominator straight l subscript 1 straight m subscript 2 minus straight l subscript 2 straight m subscript 1 end fraction.$

The equations of given lines are

straight x over straight l subscript 1 space equals space straight y over straight m subscript 1 space equals space straight z over straight n subscript 1

...(1)
and

straight x over straight l subscript 2 space equals space straight y over straight m subscript 2 space equals space straight z over straight n subscript 2

...(2)
Any line through (α, β, γ) is

fraction numerator straight x minus straight alpha over denominator straight l end fraction space equals space fraction numerator straight y minus straight beta over denominator straight m end fraction equals fraction numerator straight z minus straight gamma over denominator straight n end fraction

where l, m, n are direction-ratios of the line
Since (3) is perpendicular to (1)
∴ l l₁ + m m₁ + n n₁ = 0 ....(4)
Since (3) is perpendicular to (2)
∴ I l₂ + m m₂ + n n₂ = 0
Solving (4) and (5), we get,

fraction numerator straight l over denominator straight m subscript 1 straight n subscript 2 space minus space straight m subscript 2 straight n subscript 1 end fraction space equals space fraction numerator straight m over denominator straight n subscript 1 straight l subscript 2 minus straight n subscript 2 straight l subscript 1 end fraction space equals space fraction numerator straight n over denominator straight l subscript 1 straight m subscript 2 minus straight l subscript 2 straight m subscript 1 end fraction

∴ from (3), the equations of line are

fraction numerator straight x minus straight alpha over denominator straight m subscript 1 straight n subscript 2 minus straight m subscript 2 straight n subscript 1 end fraction space equals space fraction numerator straight y minus straight beta over denominator straight n subscript 1 straight l subscript 2 minus straight n subscript 2 straight l subscript 1 end fraction space equals space fraction numerator straight z minus straight gamma over denominator straight l subscript 1 straight m subscript 2 minus straight l subscript 2 straight m subscript 1 end fraction

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87. If the straight lines having direction cosines given by al + bm + cn = 0 and fmn + gnI + hIm = 0 are perpendicular, then show that

straight f over straight a plus straight g over straight b plus straight h over straight c space equals space 0

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Short Answer Type

88. Find the coordinates of the foot of perpendicular drawn from the point A (1, 2, 1) to the line joining B (1, 4, 6) and (5, 4, 4). Also find the perpendicular distance of A from line BC.

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89. Find the co-ordinates of the foot of the perpendicular drawn from the point A(1, 8, 4) to the line joining the points B (0, –1, 3) and C (2, – 3, – 1).

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