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Introduction To Three Dimensional Geometry

Short Answer Type

61. Prove that the four points A (1,3,0), B (- 5, 5, 2), C (- 9, - 1, 2) and D (- 3, - 3, 0) taken in order, form the vertices of a parallelogram. Do they form a rectangle also?

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62. Show that the points A (0, 7, 10), B (- 1, 6, 6), C (- 4, 9, 6) and D (- 3, 10, 10) form the vertices of a rectangle.

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63. Find points on the y-axis which are at a distance of 7 units from the point (-2, 1, 3).

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64. Verify that points A (-1, - 6, 10), B (1, - 3, 4), C (- 5, - 1, 1) and D (- 7, - 4, 7) are the vertices of a rhombus.

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65. Find a point on x-axis which is equidistant from points A (4, - 1, 2) and B (2, 0, 3).

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66.

Show that the following four points form the vertices of a square:
(a) (0, 4, 1), (2, 3, -1), (4, 5, 0) and (2, 6, 2)
(b) (- 2, 6, - 2), (0, 4, - 1), (- 2, 3, 1) and (- 4, 5, 0)

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67. Find the equation of locus of a point which moves so that its distance from point A (- 2, 1, 3) is twice its distance from the XY plane.

Solution not provided.
Ans. $straight x squared plus straight y squared minus 3 straight z squared plus 4 straight x minus 2 straight y minus 6 straight z plus 14 equals 0$

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68. Find the equation of locus of a point so that it is equidistant from the points A (1, 3, - 1) and B(4, -1, 7).

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69. Given points A (- 2, 0, 4), B (3, 2, - 1). Find the equation of locus of a point P so that PA² - PB² = 20.

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70. Find the coordinates of the point which divides the line segment formed by joining the points (- 2, 3, 5) and (1, -4, 6) in the ratio 2 : 3 internally.

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