Question and Answers Forum

All Questions      Topic List

Vector Calculus Questions

Previous in All Question      Next in All Question      

Previous in Vector Calculus      Next in Vector Calculus      

Question Number 133445 by benjo_mathlover last updated on 22/Feb/21

$$\mathrm{Show}\mathrm{that}\mathrm{the}\mathrm{plane}2\mathrm{x}-2\mathrm{y}+\mathrm{z}+12=0$$$$\mathrm{touches}\mathrm{the}\mathrm{sphere}{\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2-2\mathrm{x}-4\mathrm{y}+2\mathrm{z}-3=0$$$$\mathrm{Find}\mathrm{the}\mathrm{point}\mathrm{of}\mathrm{contact}.$$

Answered by MJS_new last updated on 22/Feb/21

$$\mathrm{sphere}:{(x-1)}^2+{(y-2)}^2+{(z+1)}^2=3^2$$$$\mathrm{center}=(1/2/-1);\mathrm{radius}=3$$$$\mathrm{distance}\mathrm{to}\mathrm{plane}=\frac{\mid 2x-2y+z+12\mid }{\sqrt{2^2+2^2+1^2}}=$$$$=\frac{\mid 2x-2y+z+12\mid }{3}=3\mathrm{for}x=1\wedge y=2\wedge z=-1$$$$\Rightarrow \mathrm{exactly}1\mathrm{intersection}$$$$\mathrm{vektor}\mathrm{\perp }\mathrm{to}\mathrm{plane}\mathrm{is}(2/-2/1)=v;\mid v\mid =3=r$$$$\mathrm{center}\mathrm{of}\mathrm{sphere}\mathrm{is}(1/2/-1)=C$$$$\mathrm{intersection}\mathrm{possible}\mathrm{at}C\pm v$$$$C-v=(-1/4/-2)\in \mathrm{plane}\Rightarrow \mathrm{point}\mathrm{of}\mathrm{contact}$$$$C+v=\left(3/0/0\right)\notin \mathrm{plane}$$

Commented by benjo_mathlover last updated on 22/Feb/21

$$\mathrm{thanks}\mathrm{prof}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com