Find an equation for each sphere that passes through the point(5,1,4) and is tangent to all three coordinate planes.

Taliyah Reyes

Taliyah Reyes

Answered question

2022-08-03

Find an equation for each sphere that passes through the point(5,1,4) and is tangent to all three coordinate planes.

Answer & Explanation

kunstboom8w

kunstboom8w

Beginner2022-08-04Added 8 answers

the general sphere equation is
( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 = R 2 ...(1)
( x 0 , y 0 , z 0 ) is the centerof sphere
since the sphere tangent to all three coordinate planes, we have:
x 0 = y 0 = z 0 = R
(1) change to:
( x R ) 2 + ( y R ) 2 + ( z R ) 2 = R
sphere passes through the point (5,1,4) , we have:
( 5 R ) 2 + ( 1 R ) 2 + ( 4 R ) 2 = R
25 10 R + R 2 + 1 2 R + R 2 + 16 8 R + R 2 = R 2
2 R 2 20 R + 32 = 0
R 2 10 R + 16 = 0
(R-2)(R-8) = 0
R=2, or R=8
the sphere equations are
( x 2 ) 2 + ( y 2 ) 2 + ( z 2 ) 2 = 2 2
or
( x 8 ) 2 + ( y 8 ) 2 + ( z 8 ) 2 = 8 2
popljuvao69

popljuvao69

Beginner2022-08-05Added 2 answers

everything answered above is totally perfect except that 25+1+16=42, and not 25+1+16=32. So, radius equals 3 and 7.

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