Short cuts in COMPLEX NUMBERS

Here goes the revision tips for a quick recollection of the important content in the chapter COMPLEX NUMBERS.

***** Equation of perpendicular bisector:

·       The equation of the perpendicular bisector of the line segment joining points   is

  

**** Important results:

If   are the points A, B and C in argand plane then 

(i)                 

(ii)               

**** Two triangles  are vertices of ABC and  vertices of DEF are similar if  

***** The equation of circle having  as the end points of the diameter is 

*** If  are fixed complex numbers then locus of a point Z satisfying is a circle having   as end points of diameter.

**** (i) The equation of circle passing through 3 points is 

(ii)if four points are concyclic
 

          must be purely real.

**** Some standard loci in the argand plane:

If z is a variable point and  are two fixed points in the argand plane then

(i)                 

Locus of z is perpendicular bisector of line segment joining 

(ii)               

 Locus of z is line segment joining 

(iii)             locus of z is ellipse

(iv)             

Locus of z is a straight line joining  but does not lie between  

(v)               

Locus of z is hyperbola.

(vi)             

 Locus of z is a circle with  as extremities of diameter.

(vii)           

Locus of z is a circle

(viii)         

Locus of z is segment of circle

(ix)            
 
Then locus of z is straight line through .

**** If are vertices of a triangle ABC then its orthocenter is 

***** The circumcenter of with  as vertices of triangle is 

Orthocenter will be

 .

*** If   are vertices of a triangle, if the triangle is equilateral then                    

    Let be circumcentre then .