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Sequences And Series

Question
CBSEENMA11014984
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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. 

Solution

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.

Some More Questions From Sequences and Series Chapter

Let P (-2, 3, -4) is a point in space. Find the foot of perpendicular from this point to the z-axis.

Let P (-2, 3, -4) is a point in space. Find the the perpendicular distance of P from XY-plane.

Name the octants in which the following points lie:
A (2, 3, 4), B (6, -3, 3), C (2, -1, -6), D (2, 2, -3), E (-1, 3, -6), F (-1, 3, 3), G (-3, -2,5) and H (-1,-2,-5).

The co-ordinates of a point A are (2, -3, 6). Write down the co-ordinates of seven points whose absolute values are the same as those of the co-ordinates of the given point.

Find the octant in which the following points lie:
A (-3,1, 2), B (-3, 1, -2), C (-3, -1,-2), D (3, -1, -2) and E (3,1, 2).

A point lies on the x-axis at a distance of 5 units towards OX'. What are its co-ordinates?

A point lies on z-axis at a distance of 3 units in the direction OZ. Write its co-ordinates.

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