Find the equation of normal to the plane at (8,5,4) :
4x−8=3y−5=0z−4=k,where k is some number.
Then
x=4k+8,y=3k+5,z=4Center of required sphere(s) is C(4k+8;3k+5;4).
Radius of required sphere(s) is
R=(4k+8−8)2+(3k+5−5)2+(4−4)2
=25k2=5∣k∣.
Sphere x2+y2+z2=1 has center (0;0;0) and radius 1.
We will have equation:
(4k+8−0)2+(3k+5−0)2+(4−0)2
=(5∣k∣+1)2
16k2+64k+64+9k2+30k+25+16
=25k2+10∣k∣+1
94k−10∣k∣=−104 k≥0: 84k=−104 No solution
k<0:94k+10k=−104
k=−1
x=4(−1)+8=4,y=3(−1)+5=2,z=4 The center of the sphere is C(4,2,4).
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