Thread: Cartesian equation
View Single Post
  #2  
Old October 15th, 2008, 02:45 PM
Soroban Soroban is offline
Super Member

  
Join Date: May 2006
Location: Lexington, MA (USA)
Posts: 7,286
Thanks: 555
Thanked 4,649 Times in 3,712 Posts
Soroban has a reputation beyond reputeSoroban has a reputation beyond reputeSoroban has a reputation beyond reputeSoroban has a reputation beyond reputeSoroban has a reputation beyond reputeSoroban has a reputation beyond reputeSoroban has a reputation beyond reputeSoroban has a reputation beyond reputeSoroban has a reputation beyond reputeSoroban has a reputation beyond reputeSoroban has a reputation beyond repute
Default
Hello, skirk34!

Quote:
Find the cartesian equation for: .| z-2+i | \:= \:\tfrac{1}{5}| z-8-3 i |

We have: .5|z - (2-i)| \:=\:|z - (8+3i)|

We want the points P(x,y) so that the distance from P to (8,3)
. . is five times the distance from P to (2,-1).


We have: .5\sqrt{(x-2)^2 + (y+1)^2} \;=\;\sqrt{(x-8)^2 + (y-3)^2}


Square both sides: .25\left[(x-2)^2 + (y+1)^2\right] \;=\;(x-8)^2 + (y-3)^2

. . which simplifies to: .24x^2 - 84x + 24y^2 + 56y \;=\;-52


Divide by 24: . x^2 - \frac{7}{2}x + y^2 + \frac{7}{3}y \;=\;-\frac{13}{6}


Complete the square: . x^2 - \frac{7}{2}x + {\color{blue}\frac{49}{36}} + y^2 + \frac{7}{3}y + {\color{red}\frac{49}{36}} \;=\;-\frac{13}{6} + {\color{blue}\frac{49}{16}} + {\color{red}\frac{49}{36}}

. . And we have: . \left(x - \frac{7}{4}\right)^2 + \left(y + \frac{7}{6}\right)^2 \;=\;\frac{325}{144}


We have a circle . . . .center \left(\frac{7}{4},\:-\frac{7}{6}\right) and radius \frac{5\sqrt{13}}{12}


But check my algebra and arithmetic . . . please!
.
Reply With Quote
The following users thank Soroban for this useful post:
Donate to MHF