find the shortest and largest distance form the point (1,2,-1) to the sphere x²+y²+z²=24
Answers
Answer:
(1,2,−1)
x
2
+y
2
+z
2
=24
length =
1
2
+2
2
+(−1)
2
=
1+4+1
=
6
Now, F(x,y,z,λ)=(x−1)
2
+(y−2)
2
+(z+1)
2
−λ(x
2
+y
2
+z
2
−24)
⇒x
2
+1−2x+y
2
+4−4y+z
2
+1+2z−λx
2
−λy
2
−24λ
=(1+λ)x
2
+(1−λ)y
2
−2x+1−4y+4+2z+1+(1−λ)z
2
+24λ
⇒
∂x
∂F
=2x+2λx−2=0 __(1)
∂y
∂F
=2y−2λy−4=0 ___(2)
∂z
∂F
=2z−2λz+2=0 ___(3)
∂λ
∂F
=−x
2
−y
2
+2y−z
2
=0 ___(4)
x+λx−1=0
x=
1+λ
1
y−λy−2=0
y=
1−λ
+2
z+λz+1=0
z=−
1−λ
1
we get ,
y=−2z,x=
2(1−y)
−y
Now, putting x ^ z values in equation (4) we get
−(−
2(1−y)
y
)
2
−y
2
−(
2
−y
)
2
+24=0
4(1−y)
2
y
2
+y
2
+
4
y
2
−24=0
y
2
+4y
2
(1−y)
2
+y
2
(1−y)
2
−24×4(1−y)
2
=0
y
2
+4y
2
(1+y
2
−2y)+y
2
(1+y
2
−2y)−96(1+y
2
−2y)=0
y
2
+4y
2
+4y
4
−8y
3
+y
2
+y
4
−2y
3
−96−96y
2
+192y=0
5y
4
−90y
2
−10y
3
+192y−96=0
hope it helps you
have a great day