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