Conic sections - equations in complex form

In earlier work we explored the polar form of complex numbers. The modulus (magnitude) is represented by .

Consider complex numbers representing points in the complex plane: z for any point P z1 for a particular point A z2 for a particular point B. z - z1 and z - z2 represent the displacements AP and BP repectively.

Consider a curve as the locus of point P. If the locus of P is determined by a distance relationship, then the modulus can be used to express the equation in complex form.
 

perpendicular bisector of AB	AP = BP
circle centre the origin and radius k	OP = k
circle centre A and radius k	AP = k
ellipse with foci A and B and major axis length 2a	AP + BP = 2a
the branch (on the same side as B) of the hyperbola with foci A and B and transverse axis length 2a	AP - BP = 2a
the branch (on the same side as A) of the hyperbola with foci A and B and transverse axis length 2a	BP - AP = 2a

• For the ellipse, 2a > AB (why?).
• For the branch of the hyperbola, 2a < AB (why?).

See also:

• Conic sections - ellipse - string method
• Conic sections - hyperbola machine

• QMaths12C Ex 3.5 p 129 2 acfkl
• QMaths12C Ex 3.6 p 134 1,2,3

To plot curves in complex form (e.g. in Winplot), convert to cartesian form:
 
 

circle centre (-6,-4) and radius 4 plot as 0=((x+6)^2+(y+4)^2)^0.5-4 or 0=(x+6)^2+(y+4)^2-4^2

ellipse foci (3,1) and (7,5) and length of major axis 10 plot as 0=((x-3)^2+(y-1)^2)^0.5+((x-7)^2+(y-5)^2)^0.5-10

branch of hyperbola foci (-5,1) and (3,-4) and length of transverse axis 8 branch on the same side as (3,-4) plot as 0=((x+5)^2+(y-1)^2)^0.5-((x-3)^2+(y+4)^2)^0.5-8

Practise the skills with this worksheet.