Questions: find an equation for the surface consisting of all points P for which the distance from P to the *-axis is twice the distance from P to the yz-plane. Identify the surface.

Solution Steps

To find the equation for the surface consisting of all points $P$ for which the distance from $P$ to the $x$-axis is twice the distance from $P$ to the $yz$-plane, we need to use the distance formulas. The distance from a point $(x, y, z)$ to the $x$-axis is $\sqrt{y^2 + z^2}$, and the distance from the point to the $yz$-plane is $|x|$. We set up the equation based on the given condition and simplify it to identify the surface.

We need to find an equation for the surface consisting of all points $P(x, y, z)$ such that the distance from $P$ to the $x$-axis is twice the distance from $P$ to the $yz$-plane.

The distance from a point $P(x, y, z)$ to the $x$-axis is given by the Euclidean distance formula: $$\text{Distance to the } x\text{-axis} = \sqrt{y^2 + z^2}$$

The distance from a point $P(x, y, z)$ to the $yz$-plane is the absolute value of the $x$-coordinate: $$\text{Distance to the } yz\text{-plane} = |x|$$

According to the problem, the distance from $P$ to the $x$-axis is twice the distance from $P$ to the $yz$-plane. Therefore, we set up the equation: $$\sqrt{y^2 + z^2} = 2|x|$$

To eliminate the square root, we square both sides of the equation: $$y^2 + z^2 = 4x^2$$

The equation $y^2 + z^2 = 4x^2$ represents a cone. This is because it is a quadratic equation in three variables where the cross-section in the $yz$-plane is a circle whose radius depends linearly on $x$.

Final Answer