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.

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.
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Solution

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Solution Steps

To find the equation for the surface consisting of all points P P for which the distance from P P to the x x -axis is twice the distance from P P to the yz yz -plane, we need to use the distance formulas. The distance from a point (x,y,z) (x, y, z) to the x x -axis is y2+z2 \sqrt{y^2 + z^2} , and the distance from the point to the yz yz -plane is x |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) P(x, y, z) such that the distance from P P to the x x -axis is twice the distance from P P to the yz yz -plane.

Step 2: Distance to the x x -axis

The distance from a point P(x,y,z) P(x, y, z) to the x x -axis is given by the Euclidean distance formula: Distance to the x-axis=y2+z2 \text{Distance to the } x\text{-axis} = \sqrt{y^2 + z^2}

Step 3: Distance to the yz yz -plane

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

Step 4: Set Up the Equation

According to the problem, the distance from P P to the x x -axis is twice the distance from P P to the yz yz -plane. Therefore, we set up the equation: y2+z2=2x \sqrt{y^2 + z^2} = 2|x|

Step 5: Square Both Sides

To eliminate the square root, we square both sides of the equation: y2+z2=4x2 y^2 + z^2 = 4x^2

Step 6: Identify the Surface

The equation y2+z2=4x2 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 yz -plane is a circle whose radius depends linearly on x x .

Final Answer

The equation for the surface is: y2+z2=4x2 \boxed{y^2 + z^2 = 4x^2} This surface is a cone.

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