Find the coordinates of a point on the y-axis | vedclass

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Question

(A) If a point lies on the $y$-axis,its $x$-coordinate and $z$-coordinate must be $0$. Let the point be $A(0, b, 0)$. The distance between $A(0, b, 0)

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$(0, 2, 0)$ and $(0, -6, 0)$

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(A) If a point lies on the $y$-axis,its $x$-coordinate and $z$-coordinate must be $0$. Let the point be $A(0, b, 0)$. The distance between $A(0, b, 0)$ and $P(3, -2, 5)$ is given as $5\sqrt{2}$. Using the distance formula,$AP = \sqrt{(3-0)^2 + (-2-b)^2 + (5-0)^2} = 5\sqrt{2}$. Squaring both sides,we get $(3)^2 + (-2-b)^2 + (5)^2 = (5\sqrt{2})^2$. $9 + (4 + 4b + b^2) + 25 = 50$. $b^2 + 4b + 38 = 50$. $b^2 + 4b - 12 = 0$. Factoring the quadratic equation: $(b + 6)(b - 2) = 0$. Thus,$b = 2$ or $b = -6$. The coordinates of the points are $(0, 2, 0)$ and $(0, -6, 0)$.

Find the coordinates of a point on the $y$-axis which is at a distance of $5\sqrt{2}$ from the point $P(3, -2, 5)$.

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