Find an equation for the set of all points equidistant

midtlinjeg 2021-08-08 Answered
Find an equation for the set of all points equidistant from the point (0,0,2) and the xy-plane.

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Expert Answer

Brittany Patton
Answered 2021-08-09 Author has 18501 answers
The given point and the plane P(0,0,2) and xy pane
It asks to find an equation for the set of all point. whereas, the distance between this equation and the point equal the distance between the same equation and xy-plane (xy-plane means z = 0). Use the distance formula \(\displaystyle\sqrt{{{\left({x}-{0}\right)}^{{2}}+{\left({y}-{0}\right)}^{{2}}+{\left({z}-{2}\right)}^{{2}}}}=\sqrt{{{\left({x}-{x}\right)}^{{2}}+{\left({y}-{y}\right)}^{{2}}+{\left({z}-{0}\right)}^{{2}}}}\)
Square both sides of an equation \(\displaystyle\sqrt{{{x}^{{2}}+{y}^{{2}}+{\left({z}-{2}\right)}^{{2}}}}=\sqrt{{{0}+{0}+{z}^{{2}}}}\)
Solve \(\displaystyle{x}^{{2}}+{y}^{{2}}+{\left({z}-{2}\right)}^{{2}}={z}^{{2}}\)
\(\displaystyle{x}^{{2}}+{y}^{{2}}+{z}^{{2}}-{4}{z}+{4}={z}^{{2}}\)
\(\displaystyle{x}^{{2}}+{y}^{{2}}-{4}{z}+{4}={0}\)
The required equation \(\displaystyle{z}={\frac{{{x}^{{2}}}}{{{4}}}}+{\frac{{{y}^{{2}}}}{{{4}}}}+{1}\)
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