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.
Transcript text: 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 $y z$-plane. Identify the surface.
Solution
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 y2+z2, 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.
Step 1: Define the Problem
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.
Step 2: Distance to the x-axis
The distance from a point P(x,y,z) to the x-axis is given by the Euclidean distance formula:
Distance to the x-axis=y2+z2
Step 3: Distance to the yz-plane
The distance from a point P(x,y,z) to the yz-plane is the absolute value of the x-coordinate:
Distance to the yz-plane=∣x∣
Step 4: Set Up the Equation
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:
y2+z2=2∣x∣
Step 5: Square Both Sides
To eliminate the square root, we square both sides of the equation:
y2+z2=4x2
Step 6: Identify the Surface
The equation y2+z2=4x2 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
The equation for the surface is:
y2+z2=4x2
This surface is a cone.