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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.

Let one of the points be P (x, y, z).
A (4, 0, 0) and B (-4, 0, 0) be given points.
Given condition is : $space space space open vertical bar PA close vertical bar space plus space open vertical bar PB close vertical bar space equals space 10$
Using distance formula, we have
$square root of left parenthesis straight x minus 4 right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared plus left parenthesis straight z minus 0 right parenthesis squared end root space plus space square root of left parenthesis straight x plus 4 right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared plus left parenthesis straight z minus 0 right parenthesis squared end root space equals space 10$
$rightwards double arrow$ $square root of straight x squared plus straight y squared plus straight z squared minus 8 straight x plus 16 end root space equals space 10 minus square root of straight x squared plus straight y squared plus straight z squared plus 8 straight x plus 16 end root$
Squaring both sides, we get
$straight x squared plus straight y squared plus straight z squared minus 8 straight x plus 16 space equals space 100 plus straight x squared plus straight y squared plus straight z squared plus 8 straight x plus 16 minus 20 square root of straight x squared plus straight y squared plus straight z squared plus 8 straight x plus 16 end root$
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Again, squaring both sides, we get
$25 space left parenthesis straight x squared plus straight y squared plus straight z squared plus 8 straight x plus 16 right parenthesis space equals space 16 straight x squared plus 625 plus 200 straight x$
$rightwards double arrow$ $9 straight x squared plus 25 straight y squared plus 25 straight z squared minus 225 equals 0$
Hence, the required locus is $9 straight x squared plus 25 straight y squared plus 25 straight z squared minus 225 equals 0$ .

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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.

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48. Find the distance between the following pairs of points:
(a) A (2, 3, 5), B (4, 3, 1)
(b) A (-3, 7, 2), B (2, 4, - 1)

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49. Show that the following sets of points are collinear.
A (1,2, 3), B (4, 0, 4), C (-2, 4, 2)

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50. Show that the following sets of points are collinear.
P (2, - 4, 1), B (4, 4, 3), C (3, 0, 2)

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