Final answer:
The surface described by the condition that the distance from any point P to the x-axis is 3 times the distance from P to the yz-plane is represented by the equation x^2/(1/9) - y^2/1 - z^2/1 = 0, which is a hyperboloid of one sheet.
Step-by-step explanation:
To find an equation for the surface consisting of all points P for which the distance from P to the x-axis is 3 times the distance from P to the yz-plane, we need to translate this description into mathematical terms and use the Cartesian coordinate system.
The distance from a point P(x, y, z) to the x-axis is given by the square root of y^2 + z^2. The distance from P to the yz-plane is simply the absolute value of the x-coordinate, which is |x|. According to the problem, the distance to the x-axis is thrice the distance to the yz-plane, which can be expressed as:
√(y^2 + z^2) = 3|x|
Squaring both sides to remove the square root gives:
y^2 + z^2 = 9x^2
Rearranging the terms:
x^2/(1/9) - y^2/1 - z^2/1 = 0
This is the standard form of a hyperboloid of one sheet. Thus, the equation describes a hyperboloid of one sheet, which is the identified surface.
The final answer in MCQ format would be: