```Question 192942
locus  problems  are  great  if  you  are  good  at  visualization

loci  around  a  point  is  a  circle  in  a  plane,  and  a  sphere  in  space  ok?

loci  of  2  points  is  a  line  perpendicular  to  connecting  line  thru  midpoint,  in  a  plane.

In  space  this  is  harder  to  describe.  visualize  a  sphere  around  one  point,  and  then  another  sphere  of  same  size  around  other point.  If  spheres  just  touch,  it  is  a  point.  If  they   merge  together  more,  it  is  circular  plane.

loci  around  a  line,  is  just  two  parallel  lines  equidistant  from  original  line,  in  a  plane.

in  space,   it  is  like  a  hollow  tube,cylindrical  surface,   with the  line  as  the  center  axis

1)  Similar  to  #  84
remember  loci  in a  plane  of  a  line  are  parallel  lines  at  equal  distances  from  line.
Your  problem  did  not  specify  how  far  apart,  so  we  will  just  be  convenient.  you  can  modify  to  suit.

original  line  is  y=x+4,  draw  a  rough  sketch.  x-y  coordinates.  y  intersect  at  4,  that  is  (0,4) point.   slope  is  +1.  IF  we  choose  a  line  parallel  to  this  but  3  units  to  right,  we  have  y=x+1.
And  the  counterpart,  3  units  to  left  is  , y=x+7

at  y=1,  our  loci  coordinates  are, (0,1)  and  (-6,1),,original (-3,1)
at  y=7,  loci,,(6,7)  and (0,7),,original  (3,7)

2)  the  midpoint  between  (3,4)  and  (3,-2)  is  (3,1)  and  line  perpendicluar  to  connecting  line  is
y=(+1)    Easier  to  just  sketch  roughly

3)loci  is  circle  of  radius  5  about  origin.  x^2+y^2=25    again  sketch  helps

4)  same  as  (3)  but  radius  =3,  x^2 +y^2 =9

have  you  studied  equation  for  circle  is  x^2+y^2=radius ^2

83)  sketch  helps,  original  line  ,,x=(-2),  is  a  vertical  line  thru  (-2,0)

loci  4  units  to  right  is  x=(+2),  and  4  units  to  left  is, x=(-6)
Again  both  are  vertical  lines,  parallel  to  original

84)  Again,  similar  to  (1)

sketching  the  original  line,  y  intercept is  (0,2),,slope(-1)  y=-x+2

loci  lines  are  parallel  and  equidistant,  but  again  we  do  not  know  how  far

lets  use  a  transfer  3  right  or  left  again
original  eqn,  y=-x+2,  to left  eqn  y=-x-1,  to  right  y=-x+5

generating  coordinates  we  find  original (2,0)  right (5,0),,  left  (-1,0)

likewise   (-2,4),  (1,4),  (-5,4)

Hopefully  this  is  close  to  your  needs```