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# help with coordinates

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The equation of a straight line AC is 2y=-3x+6. Given the point A(-2,6) and AB:BC = 1:5. Find the equation of straight line that passes through point B and perpendicular to line AC.

Feb 15, 2021

### 1+0 Answers

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Unless  we know  some  specific  value  for C, there  could  be  many answers  to  this

Here' s one thing  that would work

Construct  two   circles  with  radii  of 1   and  6   that  are centered at  ( -2,6)

This will guarantee  that  if  B lies  the  first circle and the  line, and  C lies  on the  second  circle  and the line, then  AB : BC  =1:5

Rearrangng the equation of the line we  haxe  that   y = (-3/2)x  + 3     (1)

The  equation of the  first circle  is ( x +2)^2  + ( y - 6)    = 1     (2)

Putting (1)  into (2)  and solving   for   either  x    we have  that

(x + 2)^2  +  (-3x/2  + 3  - 6)^2   = 1

x  = 2/sqrt (13)   - 2  =       [ 2 - 2sqrt (13)] /sqrt (13)

And  y  =  (-3/2) [ 2/sqrt (13)  - 2  ]  +  3  =  6 - 3/sqrt (13) =   [6 sqrt (13) - 3 ]  /sqrt (13)

A perpendicular  line at B will have a slope of  2/3

And the equation of this line  is

y =  (2/3) ( x -[ 2/sqrt/13 -2]) + (6 - 3/sqrt(13) )

See  here :  https://www.desmos.com/calculator/zu8eebvbso   Feb 15, 2021