Mathematics: Find the equations of the circles (if any) where the sphere ( x − 1 ) 2 + ( y + 3 ) 2 + ( z − 2 ) 2 = 4 intersects each coordinate plane.

1) The answers are: √16 and √25 . 2) The answers are: 4 and 5 .Step-by-step explanation:...

Find the equations of the circles (if any) where

Ответ:

21nvargas

12.03.2022

Each coordinate plane has one of the three coordinates set to zero. So, for example, all the points on the coordinate plane have the coordinate set to zero. So, to find the points belonging to the sphere and the coordinate plane , you have to solve the following system:

$\begin{cases} (x-1)^2 + (y+3)^2 + (z-2)^2 = 4\\z=0\end{cases}$

which simply means that you have to set z = 0 in the equation of the sphere:

$(x-1)^2 + (y+3)^2 + (0-2)^2 = 4 \iff (x-1)^2 + (y+3)^2 + 4 = 4 \iff (x-1)^2 + (y+3)^2 = 0$

This is not a circle, or if you prefer, it's the equation of a circle with center (1,-3) (on the xy plane) and radius 0, so it's the center itself.

Similarly, to find the intersection with the xz plane, we set y=0:

$(x-1)^2 + (0+3)^2 + (z-2)^2 = 4 \iff (x-1)^2 + 9 + (z-2)^2 = 4 \iff (x-1)^2 + (z-2)^2 = -5$

which is impossible, because the sum of two squares can't be negative.

Finally, to find the intersection with the yz plane, we set x=0:

$(0-1)^2 + (y+3)^2 + (z-2)^2 = 4 \iff 1 + (y+3)^2 + (z-2)^2 = 4 \iff (y+3)^2 + (z-2)^2 = 3$

Which is the equation of a circle with center (-3,2) (on the yz plane) and radius $\sqrt{3}$

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Find the equations of the circles (if any) where the sphere ( x − 1 ) 2 + ( y + 3 ) 2 + ( z − 2 ) 2 = 4 intersects each coordinate plane.

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