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Question
Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).
Solution
Let the point on the y-axis be Y\[\left( 0, y, 0 \right)\]which is equidistant from the points P\[\left( 3, 1, 2 \right)\]and Q \[\left( 5, 5, 2 \right)\]
Then, PY = QY
Now,
`sqrt((3 - 0)^2 + (1 - y)^2 + (2 - 0)^2) = sqrt((5 - 0)^2 + (5 - y)^2 + (2 - 0)^2)`
`=> sqrt((3)^2 + (1 - y)^2 + (2)^2) = sqrt((5)^2 + (5 - y)^2 + (2)^2)`
`=> sqrt(9 + (1 - y)^2 + 4) = sqrt(25 + (5 - y)^2 + 4)`
`=> 9 + (1 - y)^2 + 4 = 25 + (5 - y)^2 + 4`
`=> 9 + (1 - y)^2 + cancel(4) = 25 + (5 - y)^2 + cancel(4)`
`=> 9 + (1 - y)^2 = 25 + (5 - y)^2`
`=> 1 + y^2 - 2y = 25 - 9 + (5 - y)^2`
`=> 1 + y^2 - 2y = 16 + 25 + y^2 - 10y`
`=> 1 + cancel(y^2) - 2y = 41 + cancel(y^2) - 10y`
`=> - 2y = 41 - 1 - 10y`
`=> - 2y = 40 - 10y`
⇒ 8y = 40
⇒ y = `40/8`
⇒ y = 5
Thus, the required point on the y-axis is (0, 5, 0).
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