You are watching: Find an equation for the surface consisting of all points that are equidistant from the point
First, this is the graph I"ve depicted:
Some concepts to find such equation and also the equivalent graph because that the surface?
The distance from $(x,y,z)$ to the aircraft is just $|x-1|$, and the street from $(x,y,z)$ come $(-1,0,0)$ is $sqrt(x+1)^2+y^2+z^2$. Setup the 2 equal to eah other, us get
$$(x-1)^2 = (x+1)^2 + y^2 + z^2$$$$-4x = y^2 + z^2$$$$x = -frac14(y^2+z^2)$$
If $z$ is not considered standard a 2D meaning of parabola deserve to be obtained by setup equal ranges as:
$$x = -fracy^24 $$
So through z thought about it i do not care a 3D generalized surface of transformation as a paraboloid rotated around $x$ axis vertically up :
$$ (x,y,z) = (-2,2,2)$$
$$x = -fracy^2+z^24 $$
$ yz $ airplane is horizontal, $x$ axis is vertical as shown.
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