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.
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 $P$ have coordinates $(x,y,z)$. Then the distance from $P$ to the $x$-axis is $|y|$, and the distance from $P$ to the $yz$-plane is $|x|$. Therefore, we have the equation $$2|y| = |x|.$$ Squaring both sides, we get $$4y^2 = x^2.$$ Hence, the surface consists of all points $(x,y,z)$ satisfying $$4y^2 = x^2,$$ or equivalently, $$4y^2 = x^2 + 0^2 + z^2.$$ This is the equation of a double cone with its axis along the $y$-axis. To get the equation in the desired form, we can divide both sides by $4$ to get $$\frac{x^2}{4} - \frac{y^2}{4} - \frac{z^2}{4} = 0,$$ which is equivalent to $$4x^2 = y^2 + z^2.$$ Finally, multiplying both sides by $4$, we obtain the equation $$4x^2 = y^2 + z^2,$$ which matches choice $\boxed{\textbf{(B)}}$.

Find an Equation for the Surface Consisting of All Points PPP

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