Locus of $P(4\cos t, 4\sin t, 4\sin t)$ in space.

Question

A variable point $P(4 \cos t, 4 \sin t, 4 \sin t)$ moves in space, now which of the following is true

1. Point $P$ moves on a plane $ax + by + cz + d = 0$
2. Point $P$ traces a circle
3. Area enclosed by P is $16\sqrt{2}\pi$
4. Point P cannot lie on a fixed circle

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I wrote the equation as $ 2x^2 + y^2 + z^2 = 32$. But I cannot proceed any further. Or is there any other method?

Answer 1

As you can see the locus is not a circle or a plane, so options 1, 2, 3 are wrong. Also the volume of an ellipsoid with equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ is $$\frac{4}{3}\pi abc$$. Hence volume of this ellipsoid will be $16\sqrt{2}\pi$