Find the equation of normal to the plane at ( 8 , 5 , 4 ) (8,5,4) ( 8 , 5 , 4 ) :
x − 8 4 = y − 5 3 = z − 4 0 = k , \frac{x-8}{4}=\frac{y-5}{3}=\frac{z-4}{0}=k, 4 x − 8 = 3 y − 5 = 0 z − 4 = k , where k k k is some number.
Then
x = 4 k + 8 , y = 3 k + 5 , z = 4 x=4k+8, y=3k+5, z=4 x = 4 k + 8 , y = 3 k + 5 , z = 4 Center of required sphere(s) is C ( 4 k + 8 ; 3 k + 5 ; 4 ) . C(4k+8;3k+5;4). C ( 4 k + 8 ; 3 k + 5 ; 4 ) .
Radius of required sphere(s) is
R = ( 4 k + 8 − 8 ) 2 + ( 3 k + 5 − 5 ) 2 + ( 4 − 4 ) 2 R=\sqrt{(4k+8-8)^2+(3k+5-5)^2+(4-4)^2} R = ( 4 k + 8 − 8 ) 2 + ( 3 k + 5 − 5 ) 2 + ( 4 − 4 ) 2
= 25 k 2 = 5 ∣ k ∣ . =\sqrt{25k^2}=5|k|. = 25 k 2 = 5∣ k ∣.
Sphere x 2 + y 2 + z 2 = 1 x^2+y^2+z^2=1 x 2 + y 2 + z 2 = 1 has center ( 0 ; 0 ; 0 ) (0;0;0) ( 0 ; 0 ; 0 ) and radius 1. 1. 1.
We will have equation:
( 4 k + 8 − 0 ) 2 + ( 3 k + 5 − 0 ) 2 + ( 4 − 0 ) 2 (4k+8-0)^2+(3k+5-0)^2+(4-0)^2 ( 4 k + 8 − 0 ) 2 + ( 3 k + 5 − 0 ) 2 + ( 4 − 0 ) 2
= ( 5 ∣ k ∣ + 1 ) 2 =(5|k|+1)^2 = ( 5∣ k ∣ + 1 ) 2
16 k 2 + 64 k + 64 + 9 k 2 + 30 k + 25 + 16 16k^2+64k+64+9k^2+30k+25+16 16 k 2 + 64 k + 64 + 9 k 2 + 30 k + 25 + 16
= 25 k 2 + 10 ∣ k ∣ + 1 =25k^2+10|k|+1 = 25 k 2 + 10∣ k ∣ + 1
94 k − 10 ∣ k ∣ = − 104 94k-10|k|=-104 94 k − 10∣ k ∣ = − 104 k ≥ 0 : k\ge0: k ≥ 0 : 84 k = − 104 84k=-104 84 k = − 104 No solution
k < 0 : 94 k + 10 k = − 104 k<0: 94k+10k=-104 k < 0 : 94 k + 10 k = − 104
k = − 1 k=-1 k = − 1
x = 4 ( − 1 ) + 8 = 4 , y = 3 ( − 1 ) + 5 = 2 , z = 4 x=4(-1)+8=4, y=3(-1)+5=2, z=4 x = 4 ( − 1 ) + 8 = 4 , y = 3 ( − 1 ) + 5 = 2 , z = 4 The center of the sphere is C ( 4 , 2 , 4 ) . C(4, 2, 4). C ( 4 , 2 , 4 ) .
Comments