Given plane equation is (x+y+2z-2)=0, sphere equation is (x²+y²+z²+3x+3y-z)=1
x2+2⋅23x+49−49+y2+2⋅23y+49−49+
+z2−2⋅21z+41−41=1
(x+23)2+(y+23)2+(z−21)2=423
Centre(−u,−v,−w)=(−23,−23,21)
Radius r=423=223
Length of the perpendicular from (−23,−23,21) to the plane (x+y+2z-2) =0 is
OM=a2+b2+c2∣∣ax1+by1+cz1+d∣∣ Here, a=1,b=1,c=2,d=−2,x1=−23,y1=−23,z1=21
OM=12+12+22∣∣−23−23+2⋅21−2∣∣=326
AM2=OA2−OM2=(223)2−(326)2=1237 The equation of the axis of the cylinder is
lx−α=my−β=nz−γ
Here, α=−23,β=−23,γ=21,l=1,m=1,n=2.
1x+23=1y+23=2z−21
l2+m2+n2=6
Radius r∗=AM=1237=6111 The equation of Right Circular Cylinder is,
[n(y−β)−m(z−γ)]2+[l(z−γ)−n(x−α)]2++[m(x−α)−l(y−β)]2=(r∗)2(l2+m2+n2)
[2(y+23)−1(z−21)]2+[1(z−21)−2(x+23)]2++[1(x+23)−1(y+23)]2=(6111)2(6)
4y2+z2+449−4yz+14y−7z+z2+4x2+449−−4xz−7z+14x+x2−2xy+y2=237
5x2+5y2+2z2−2xy−4xz−4yz+14x+14y−14z+6=0
Comments
Leave a comment