Cylinder

Author Bhawna asked on 2019-07-22 13:26:00

 

Let P(x1, y1, z1) be any point on the cylinder.

 The generator through P is parallel to the line, x /2 = y/ -1 = z/ 2

 Therefore the equation of the generator through P  are  , x − x1/ 2 = y − y1/- 1 = z − z1/ 2 = t

Coordinates of any point on this line are (x1+2t, y1-t, z1+2t). For some t, this point lies on guiding curve   x+y+z=1 , X2+y2+z2=4 

  (x1 + 2t) + (y1-t)+ (z1+2t)=1 --i,   (x1 + 2t)2 + (y1-t)2+ (z1+2t)2=4--ii

    x1  +y1+ z1+3t=1

    x1  +y1+ z1-1 =3t

    (x1  +y1+ z1-1)/3 =t

Put the value of t in equation ii

    Also (x1 + 2t)2 + (y1-t)2+ (z1+2t)2=4

             {x1 + 2(x1  +y1+ z1-1)/3) }2 + (y1-(x1  +y1+ z1-1)/3)2+ (z1+2(x1  +y1+ z1-1)/3)2=4

             {(3x1 + 2x1  +2y1+ 2z1-2)/3 }2 + {(3y1-x1  -y1-z1+1)/3}2+ {(3z1+2x1  +2y1+2 z1-2)/3}2=4

             {(5x1 +2y1+ 2z1-2)/3 }2 + {(2y1-x1  -z1+1)/3}2+ {(5z1+2x1  +2y1 -2)/3}2=4

 

Hence, locus of P is, {(5x +2y+ 2z-2)/3 }2 + {(2y-x  -z+1)/3}2+ {(5z+2x +2y -2)/3}2=4