Finding and Identifying a Quadric Surface Equation
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1. a) (5 pts) Let $S$ be the surface consisting of all points $Q$ satisfying the following property: The distance from $Q$ to $x$-axis is twice the distance from $Q$ to the $yz$-plane. Write the equation of $S$ in the box below.
Solution: Let $Q = (x, y, z)$. Then the distance from $Q$ to the $x$-axis is given by $d_1 = \sqrt{y^2 + z^2}$ whereas the distance from $Q$ to the $yz$-plane is given by $d_2 = |x|$. From the given condition $d_1 = 2d_2$ implying $\sqrt{y^2 + z^2} = 2|x|$, or equivalently by squaring both sides $4x^2 = y^2 + z^2$.
$$4x^2 = y^2 + z^2$$
b) (5 pts) Identify the surface is $S$ in part (a) as in the classification of quadric surfaces and draw it in the coordinate system below.
Solution: The surface $S$ is a cone according to the classification of quadric surfaces.
This question includes visual content: The image contains two visual elements: 1) A grading table at the top with columns Q1 (10 pts), Q2 (15 pts), Q3 (15 pts), Q4 (20 pts), and a total of 60 pts. 2) A 3D coordinate system sketch showing a double-napped cone opening along the x-axis. Labels for x, y, and z axes are present, with the cone centered at the origin $(0,0,0)$. The circular cross-sections of the cone are shown in perspective.
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Step by Step Written Solution
In this problem, we are tasked with finding the equation of a surface S based on a distance relationship and then identifying and sketching that surface.
Defining Surface S
Let's start with part a. Define a general point Q on the surface with coordinates x, y, and z.
The problem states two distance conditions. First, the distance from Q to the x-axis, let's call it d sub one, is the square root of y squared plus z squared.
Second, the distance from Q to the y-z plane, let's call it d sub two, is simply the absolute value of the x coordinate.
The condition given is that d sub one is twice d sub two. This gives us our initial equation.
Substituting the expressions for d sub one and d sub two, we get the square root of y squared plus z squared equals two times the absolute value of x.
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