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Mathematics : Introduction To Three Dimensional Geometry

Long Answer Type

121. ABCD is quadrilateral, the co-ordinates of whose vertices are A (-1, 3, 2), B (3, -5, 4), C (5, -1, 0) and (-3, 1, 6). Show that mid-points of sides AB, BC, CD and DA form a parallelogram.

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Short Answer Type

122. In this figure below, the co-ordinates of point G in space are (a, b, c). Find the co-ordinates of the remaining vertices of the cuboid.

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123.

In the figure above, the co-ordinates of point G in space are (6, 4, 5). Find the co-ordinates of points O, A, B, C, D, E and F.

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Long Answer Type

124. If P(x₁, y₁, z₁) and Q(x₂,y₂,z₂) be the end points of a line segment PQ and let R be a point on PQ dividing PQ in the ratio m : n internally, then prove that the co-ordinates of R are:

space space open parentheses fraction numerator mx subscript 2 plus nx subscript 1 over denominator straight m plus straight n end fraction comma space fraction numerator my subscript 2 plus ny subscript 1 over denominator straight m plus straight n end fraction comma space fraction numerator mz subscript 2 plus nz subscript 1 over denominator straight m plus straight n end fraction close parentheses

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Short Answer Type

125.

If $straight A left parenthesis straight x subscript 1 comma space straight y subscript 1 comma space straight z subscript 1 right parenthesis comma space straight B left parenthesis straight x subscript 2 comma space straight y subscript 2 comma space straight z subscript 2 right parenthesis space and space straight C left parenthesis straight x subscript 3 comma space straight y subscript 3 comma space straight z subscript 3 right parenthesis$ are the vertices of triangle ABC, then prove that its centroid G is $open parentheses fraction numerator straight x subscript 1 plus straight x subscript 2 plus straight x subscript 3 over denominator 3 end fraction comma space fraction numerator straight y subscript 1 plus straight y subscript 2 plus straight y subscript 3 over denominator 3 end fraction comma space fraction numerator straight z subscript 1 plus straight z subscript 2 plus straight z subscript 3 over denominator 3 end fraction close parentheses.$

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Long Answer Type

126. Distance between two points: If P(x₁, y₁ z₁) and Q(x₂, y₂, z₂) are two points in space, then prove that distance

PQ space equals space square root of left parenthesis straight x subscript 2 minus straight x subscript 1 right parenthesis squared plus left parenthesis straight y subscript 2 minus straight y subscript 1 right parenthesis squared plus left parenthesis straight z subscript 2 minus straight z subscript 1 right parenthesis squared end root

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Multiple Choice Questions

127.

The number of common tangent to the circles x²+y²-4x-6y-12=0 and x²+y²+6x+18y+26 = 0 is

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128.

The angle between the lines whose direction cosines satisfy the equations l +m+n=0 and l² = m²+n² is

π/3
π/4
π/6
π/6

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129.

If PS is the median of the triangle with vertices P(2,1), Q(6,-1) and R (7,3), then equation of the line passing through (1,-1) and parallel to PS is

4x-7y - 11 =0
2x+9y+7=0
4x+7y+3 = 0
4x+7y+3 = 0

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130.

Let a,b,c and d be non-zero numbers. If the point of intersection of the lines 4ax +2ay +c= 0 and 5bx +2by +d = 0 lies in the fourth quadrant and is equidistant from the two axes, then

2bc-3ad =0
2bc+3ad =0
2ad-3bc =0
2ad-3bc =0

252 Views