Find the coordinates of a point on y-axis which are at a distance of 5√2 from the point P (3, –2, 5).

Let the point on y-axis be A (0, y, 0).

Then, it is given that the distance between the points A (0, y, 0) and P (3, -2, 5) is \( 5\sqrt{2}\).

Now, by using distance formula,

We know that the distance between two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) is given by

Distance of PQ = 

\( \sqrt{(x^2 – x^1)^2 + (y^2 – y^1)^2 + (z^2 – z^1)^2}\)

So, the distance between the points A (0, y, 0) and P (3, -2, 5) is given by

= \(\sqrt{(3-0)^2 + (-2-y)^2 + (5-0)^2}\)

= \( \sqrt{3^2 + (-2-y)^2 + 5^2}\)

= \(\sqrt{(-2-y)^2 + 9 + 25}\)

\(5 \sqrt{2}\) = \(\sqrt{(-2-y)^2 + 34}\)

Squaring on both the sides, we get

(-2 -y)² + 34 = 25 × 2

(-2 -y)² = 50 – 34

4 + y² + (2 × -2 × -y) = 16

y² + 4y + 4 -16 = 0

y² + 4y – 12 = 0

y² + 6y – 2y – 12 = 0

y (y + 6) – 2 (y + 6) = 0

(y + 6) (y – 2) = 0

y = -6, y = 2

The points (0, 2, 0) and (0, -6, 0) are the required points on the y-axis.