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$A=kx$ is a directly proportional function,where $A^2=B^2+C^2$.Does it necessarily mean $B$ and $C$ both vary directly with respect to $x$? If not, under what condition is this possible?

Thank you in advance for your explanation.

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No. If $B(x)$ is an arbitrary function bounded between $0$ and $\sqrt{kx}$, we can simply let $C(x) = \sqrt{kx - B(x)^2}$, and the conditions are satisfied. Your condition will need to restrict $B(x)$ to be linear if you wish for $B$ and $C$ to be proportional to $x$.

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