Complex Numbers - Cartesian Form
From WJEC FP1 (June 2005) - Question 1
_______________________________________________________________
The complex number z is represented by the point P on an Argand diagram.
Given that
| z + 1 | = 2 | z - 2i |
find, in its simplest form, the Cartesian equation of the locus of P. [5]
_______________________________________________________________
Firstly z is a complex number and can be written in the form x + iy. Since z is represented by the point P then the equation of the locus of P is an equation involving both x and y. So if we substitiute x + iy wherever there is a z then we have
| x + iy + 1 | = 2 | x + iy - 2i |
We must now collect the 'real' and 'imaginary' terms on each side together, which we will eventually square to remove the modulus on each side. Therefore
| (x + 1) + iy | = 2 | x + (iy - 2i) |
The definition of modulus in this instance is the square root of the sum of the real parts sqaured and the imaginary parts squared. So we have
√( (x + 1)² + y² ) = 2 √( x² + ( y - 2)² )
(Note: We are taking the co-efficients of the imaginary parts here so iis not included)
Expanding the brackets
√( (x² + 2x + 1) + y² ) = 2 √( x² + ( y² - 4y + 4) )
There is a square root on both sides of the equation, so I will square out both sides to remove them.
( x² + 2x + 1 ) + y² = 4 ( x² + ( y² - 4y + 4 ) )
Rearranging the right hand side by expanding out the brackets
x² + 2x + 1 + y² = 4x² + 4y² - 16y + 16
Collecting the like terms
0 = 4x² + 4y² - 16y + 16 - x² - 2x - 1 - y²
Then simplifying gives us
3x² + 3y² - 16y - 2x + 15 = 0
Comments (2)
Steph Richards said
at 5:58 pm on Dec 4, 2008
Good work Tom. Please tell me how you wrote the mathematical text.
Tom Newton said
at 11:17 pm on Jan 8, 2009
The '²' is done by holding down the ALT key on your keyboard and pressing the code 0178 on the numpad. Likewise for '³' you use ALT+0179. Alternatively you could go to the 'Run' option on the Start menu of Windows and type 'charmap'. This opens up the list of all characters available for a particular font, and is how I got the square root sign.
You don't have permission to comment on this page.