Mathematics

LIKE TO JOIN ON FACEBOOK ( click here ) Equation of Ellipse: Ellipse is locus of a point whose sum of distance from two fixed points is constant.                                                   OR Ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one focal point to any point of the curve and then back to other focal point has the same length for every point of the curve. |z - z1| + |z - z2| = 2a where, 2a > |z1 - z2| (if 2a = |z1 - z2| , equation represent a line segment and if 2a < |z1 - z2|, there is no value of z i.e. sum of distance from two fixed points cannot be lesser than distance between those two fixed points) Focii -> z1 and z2 2a -> Major axis Ques: |z - 1| + |z + 1| = 4, find the locus of z ? Solution: Locus of z is ellipse, Focii -> -1 and 1 or (...

LIKE TO JOIN ON FACEBOOK ( click here ) 1.   a) if k = 1 , then the locus of 'z' is a straight line i.e. perpendicular bisector of line joining the point and . b) if  , then the locus of 'z' is circle By definition of circle, the ratio of distance from two points of a point is constant then the locus of that point is circle. c) if k = 0, the locus of z is a point which is equal to . Ques:  or,   ,  and, since  , i.e.ratio of distance of 'z' from i (0,1) and -i (0,-1) is 1/3 by section formula,  ( internally,   )  ( externally,   ) The points gained from section formula will always be the end points of the diameter of the circle . 2.   or,    or,   The locus is an arc of a circle  and line joining z1 and z2 as a chord subtending angle   and excluding the points z1 and z2. [ angle should be measured along anti-clockwise direction from ...

LIKE TO JOIN ON FACEBOOK ( click here ) Standard results in Complex number (Argand Plane) 1. Section Formula :- 2.  Centroid:- centroid,   3. Equation of a Straight line :- a) Parametric Form:- From triangle OBP or,  b)    or  =>   or   is Purely Real  ........................ (i) And, therefore condition of collinearity of  because if    are collinear then area form by these points must be 0. General Equation of a Straight Line: Where 'a' is a complex number and 'b' is a Purely Real Number. a - complex number, b - Purely Real Number Proof:- From (i) we have , multiply  by 'i' on both sides, Suppose,    and  and   ( since,   ) And,     = b (let) (We know that,   =  Purely Real ) therefore,   = Purely Real (  'b' ) then equation becomes , Where...