Find an equation for the surface consisting of all points $P$ for which the distance from $P$ to the $x$-axis is two times the distance from $P$ to the $y z$-plane. Select the correct answer.
A) $2 x ^ { 2 } = y ^ { 2 } - 2 z ^ { 2 }$
B) $4 x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$
C) $2 x ^ { 2 } = y ^ { 2 } + 2 z$
D) $x ^ { 2 } - y ^ { 2 } - z ^ { 2 } = 2$
E) $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$

Question

Quizplus · Accepted Answer

Let $P=(x,y,z)$ be any point on this surface. Then, we have:
$$\text{Distance from }P\text{ to x-axis}=\left|y\right|$$
$$\text{Distance from }P\text{ to }yz\text{-plane}=\left|x\right|$$
$$\because\text{Distance from }P\text{ to x-axis} = 2\cdot\text{Distance from }P\text{ to }yz\text{-plane}$$
$$\Rightarrow\left|y\right|=2\cdot\left|x\right|$$
$$\Rightarrow 4x^2=y^2+z^2$$
Therefore, the correct answer is B.

Find an Equation for the Surface Consisting of All Points PPP

Find an Equation for the Surface Consisting of All Points $P$