Describe the given set with a single equation or with a pair of equations.
The set of points in space that lie 2 units from the point (0,0,1) and, at the same time, 2 units from the point (0,0,-1).
I wrote distance equations for both set them equal to each other and got z=0. Then, I plugged in 0 for z and got x^2+y^2=3. I'm just not sure what the answer would be. Is it z=0 or x^2+y^2=3?
points equidistant from a point in 3D are the surface of a sphere.
So, your two constraints will be the intersection of two spheres, which is a circle.
Naturally, the intersection will lie in the x-y plane, since the centers of the spheres are both on the z-axis.
Your circle is indeed x^2+y^2 = 3. It just happens to lie in the plane z=0. Technically, x^2+y^2=3 is a cylinder centered on the z-axis.
Great job on your work so far! Let's break down the problem and determine the equation(s) that describe the given set.
We are looking for a set of points in space that are equidistant from the points (0,0,1) and (0,0,-1). The distance between a point (x,y,z) and a given point (a,b,c) can be found using the distance formula:
d = √((x-a)^2 + (y-b)^2 + (z-c)^2)
To find the set of points that are equidistant from two points, we'll set the distances equal to each other.
So, the distance from a point (x, y, z) to (0, 0, 1) is given by:
d1 = √((x-0)^2 + (y-0)^2 + (z-1)^2)
= √(x^2 + y^2 + (z-1)^2)
And the distance from the same point (x, y, z) to (0, 0, -1) is given by:
d2 = √((x-0)^2 + (y-0)^2 + (z-(-1))^2)
= √(x^2 + y^2 + (z+1)^2)
Setting these two distances equal to each other, we have:
√(x^2 + y^2 + (z-1)^2) = √(x^2 + y^2 + (z+1)^2)
To get rid of the square root, we can square both sides of the equation:
x^2 + y^2 + (z-1)^2 = x^2 + y^2 + (z+1)^2
Simplifying the equation, we have:
(z-1)^2 = (z+1)^2
Expanding both sides of the equation:
z^2 - 2z + 1 = z^2 + 2z + 1
Simplifying the equation further, we can cancel out the z^2 terms:
-2z = 2z
Dividing both sides by 2, we get:
z = 0
So, the equation that describes the set of points in space that are 2 units from both (0, 0, 1) and (0, 0, -1) is z = 0.
To summarize, the set can be described by the equation z = 0. The equation x^2 + y^2 = 3 that you obtained is not necessary to define this particular set since it does not involve the z-coordinate.