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(C) The line $OP$ passes through the origin $O(0, 0, 0)$ and has direction ratios $3, -2, 6$. Thus, the coordinates of any point $P$ on this line can

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$(27, -18, 54)$

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(C) The line $OP$ passes through the origin $O(0, 0, 0)$ and has direction ratios $3, -2, 6$. Thus, the coordinates of any point $P$ on this line can be written as $(3\lambda, -2\lambda, 6\lambda)$ for some scalar $\lambda$. The distance of point $P$ from the origin $O$ is given by $|OP| = \sqrt{(3\lambda)^2 + (-2\lambda)^2 + (6\lambda)^2}$. Given that $|OP| = 63$, we have: $\sqrt{9\lambda^2 + 4\lambda^2 + 36\lambda^2} = 63$ $\sqrt{49\lambda^2} = 63$ $7|\lambda| = 63$ $|\lambda| = 9$ Taking $\lambda = 9$, the coordinates of $P$ are $(3(9), -2(9), 6(9)) = (27, -18, 54)$. Taking $\lambda = -9$, the coordinates of $P$ are $(-27, 18, -54)$. Comparing with the given options, the correct coordinates are $(27, -18, 54)$.

Suppose the distance of a point $P$ from the origin $O$ is $63$. If the direction ratios of the line $OP$ are $3, -2$ and $6$, then the coordinates of the point $P$ are:

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