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Question
Find the equation of the 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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Solution
Let the coordinates of point P be (x, y, z).
Given points A(4, 0, 0) and B(−4, 0, 0) such that
PA + PB = 10
`sqrt((x - 4)^2 + (y - 0)^2 + (z - 0)^2) + sqrt((x + 4)^2 + (y + 0)^2 + (z - 0)^2)` = 10
or `sqrt(x^2 + y^2 + z^2 - 8x + 16) = 10 - sqrt(x^2 + y^2 + z^2 + 8x + 16)`
On squaring both sides,
`x^2 + y^2 + z^2 - 8x + 16 = 100 + (x^2 + y^2 + z^2 + 8x + 16) - 20 sqrt(x^2 + y^2 + z^2 + 8x + 16)`
−16x −100 = −20`sqrt(x^2 + y^2 + z^2 + 8x + 16)`
= 4x + 25 = `5sqrt(x^2 + y^2 + z^2 + 8x + 16)`
On squaring both sides again
= (4x + 25)2 = 25(x2 + y2 + z2 + 8x + 16)
= 16x2 + 200x + 625 = 25x2 + 25y2 + 25z2 + 200x + 400
= 9x2 + 25y2 + 25z2 - 225 = 0
Hence: Required equation 9x2 + 25y2 + 25z2 - 225 = 0
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