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All the points equidistant to a given point would form?
A circle.
Where do all the points in space equidistant from a given point lie?
I'm not sure, but I would imagine they would be 360O around the point and only in the same plane.
What is a space figure who set of all points on the surface are equidistant from the center?
A sphere would fit the given description.
How do you place four points equidistant from each other?
To place four points equidistant from each other, you would need to arrange them in the shape of a perfect square. This means that each point would be the same distance away from the other three points, forming equal sides of the square. The distance between each point can be calculated using the Pythagorean theorem if the coordinates of the points are known.
Are the directrix and focus different distances from a given point on a parabola?
One definition of a parabola is the set of points that are equidistant from a given line called the directrix and a given point called the focus. So, no. The distances are not different, they are the same. The distance between the directrix and a given point on the parabola will always be the same as the distance between that same point on the parabola and the focus. Any point where those two distances are equal would be on the parabola somewhere and all the points where those two distances are different would not be on the parabola. Note that the distance from a point to the directrix is definied as the perpendicular distance (also known as the minimum distance).
