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Introduction To Three Dimensional Geometry

Short Answer Type

41. Find the value of x, so that the point (6, 5, -3) is at a distance of 13 units from the point (x, -7, 0).

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42. Find the co-ordinates of a point on the z-axis which is equidistant from points A (3, 2, 1) and B (5, 2, 5).

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43.

Find points on x - axis which are at a distance of $square root of 29$ units from point A (1, 2, 3).

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44. Find the equation of the set of points which are equidistant from the points (1, 2, 3) and B (3, 2, -1).

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Long Answer Type

45. Find the equation of set of points P the sum of whose distances from A (4, 0, 0) and B (-4, 0, 0) is equal to 10.

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Short Answer Type

46. If A and B be the points (3, 4, 5) and (-1, 3, - 7) respectively, find the equation of set of points P such that PA² + PB² = 2k². where k is a constant.

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47. Find the equation of the locus of a point P or the equation of set of points P which moves so that its distance from A (1, 2, 3) is four times its distance from the YZ-plane.

Let point P be (x, y, z) and A (1, 2, 3) be the given point. The given condition for locus is:
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Squaring both sides, we get
$space space left parenthesis straight x minus 1 right parenthesis squared plus left parenthesis straight y minus 2 right parenthesis squared plus left parenthesis straight z minus 3 right parenthesis squared space equals space 16 straight x squared$
Hence, locus of P is $15 straight x squared minus straight y squared minus straight z squared plus 2 straight x plus 4 straight y plus 6 straight z minus 14 equals 0$