Argand Plane

We have,

%5Cbegin%7Bvmatrix%7D%20%5Csin%5C%2C%20x%20%26%5Ccos%5C%2C%20x%20%26%20%5Ccos%5C%2C%20x%5C%5C%20%5Ccos%5C%2C%20x%20%26%20%5Csin%5C%2C%20x%20%26%5Ccos%5C%2Cx%5C%5C%20%5Ccos%5C%2C%20x%20%26%5Ccos%5C%2Cx%20%26%5Csin%5C%2C%20x%5Cend%7Bvmatrix%7D%3D0

Applying

C_%7B1%7D%20%5Cto%20C_%7B1%7D%2BC_%7B2%7D%2BC_%7B3%7D

, we get

%5Cbegin%7Bvmatrix%7D2%5C%2C%20%5Ccos%5C%2C%20x%20%2B%20%5Csin%5C%2C%20x%20%26%5Ccos%5C%2C%20x%20%26%5Ccos%5C%2C%20x%5C%5C%202%5C%2C%20%5Ccos%5C%2C%20x%20%2B%20%5Csin%5C%2C%20x%20%26%5Csin%5C%2C%20x%20%26%20%5Ccos%5C%2C%20x%5C%5C%202%5C%2C%20%5Ccos%5C%2C%20x%20%2B%5Csin%5C%2C%20x%20%26%5Ccos%5C%2C%20x%20%26%5Csin%5C%2Cx%5Cend%7Bvmatrix%7D%3D0

Taking

(2%5C%2C%20%5Ccos%5C%2C%20x%20%2B%20%5Csin%5C%2C%20x)

common from

C_1

, we get

%5Cleft(2%5C%2C%20%5Ccos%5C%2C%20x%20%2B%20%5Csin%5C%2Cx%5Cright)%5Cbegin%7Bvmatrix%7D1%26%20%5Ccos%5C%2C%20x%20%26%5Ccos%5C%2Cx%5C%5C%201%26%5Csin%5C%2C%20x%20%26%20%5Ccos%5C%2Cx%5C%5C%201%26%5Ccos%5C%2C%20x%20%26%5Csin%5C%2Cx%5Cend%7Bvmatrix%7D%3D0

Applying

R_%7B2%7D%20%5Cto%20R_%7B2%7D%20-R_%7B1%7D

and

R_%7B3%7D%20%5Cto%20R_%7B1%7D

, we get

%5Cleft(2%5C%2C%20%5Ccos%5C%2C%20x%20%2B%5Csin%5C%2Cx%5Cright)%5Cbegin%7Bvmatrix%7D1%26%5Ccos%5C%2C%20x%20%26%5Ccos%5C%2Cx%5C%5C%200%26%5Csin%5C%2Cx-%5Ccos%5C%2Cx%260%5C%5C%200%260%26%5Csin%5C%2Cx-%5Ccos%5C%2Cx%5Cend%7Bvmatrix%7D%3D0

Expanding along

C_%7B1%7D

, we get

%5Cleft(2%5C%2C%20%5Ccos%20%5C%2Cx%20%2B%20%5Csin%20%5C%2Cx%5Cright)%20%5Cleft%5B1%20%5Cleft(%5Csin%5C%2C%20x%20-%20%5Ccos%20%5C%2Cx%5Cright)%5E%7B2%7D%5Cright%5D%20%3D%200

%5CRightarrow%20%5Cleft(2%20%5C%2C%5Ccos%20%5C%2Cx%20%2B%20%5Csin%20%5C%2Cx%5Cright)%20%5Cleft(%5Csin%20%5C%2Cx%20-%20%5Ccos%20%5C%2Cx%5Cright)%5E%7B2%7D%3D0

Now, if

2%5Ccos%5C%2Cx%20%2B%20%5Csin%5C%2Cx%20%3D%200

, then

2%5C%2C%5Ccos%20%5C%2Cx%20%3D%20-%20%5Csin%5C%2C%20x

%5CRightarrow%20%5Ctan%5C%2Cx%3D-2

But here

-%5Cfrac%7B%5Cpi%7D%7B4%7D%20%5Cle%20x%20%5Cle%20%5Cfrac%7B%5Cpi%7D%7B4%7D

, we have

-1%20%5Cle%20%5Ctan%5C%2Cx%20%5Cle%201

, so no solution possible.
or if

%5Cleft(%5Csin%5C%2C%20x-%5Ccos%5C%2Cx%5Cright)%5E%7B2%7D%3D0

, then

%5Csin%5C%2Cx%20%3D%5Ccos%5C%2Cx

%5CRightarrow%20%5Ctan%5C%2Cx%3D1%3D%5Ctan%20%5Cfrac%7B%5Cpi%7D%7B4%7D

%5CRightarrow%20x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D

So, only one distinct real root exist.

We have,

%5Carg%20%5Cleft(%5Cfrac%7Bz%2Bi%7D%7Bz-i%7D%5Cright)%3D%5Cfrac%7B%5Cpi%7D%7B4%7D

Let

Z%20%3D%20x%20%2B%20i%20y

%5Ctherefore%20%5Cfrac%7B%20z%20%2B%20i%20%7D%7B%20z%20-%20i%20%7D%20%3D%5Cfrac%7B%20x%20%2B%20i%20y%2B%20i%20%7D%7B%20x%20%2B%20i%20y-%20i%20%7D

%3D%5Cfrac%7B%20x%20%2B(%20y%20%2B%20l%20)%20i%20%7D%7B%20x%20%2B(%20y%20-%20l%20)%20i%20%7D

%3D%5Cfrac%7B(%20x%20%2B(%20y%20%2B%20l%20)%20i%20)(%20x%20-(%20y%20-%20l%20)%20i%20)%7D%7B(%20x%20%2B(y-1)%20i%20)(%20x%20-(%20y%20-%20l%20)%20i%20)%7D

%3D%5Cfrac%7B%20x%20%5E%7B2%7D%2B%20y%20%5E%7B2%7D-1%2B(-%20x%20y%2B%20x%20%2B%20x%20y%20%2B%20x%20)%20i%20%7D%7B%20x%20%5E%7B2%7D%2B(%20y%20-1)%5E%7B2%7D%7D

%3D%5Cfrac%7B%20x%20%5E%7B2%7D%2B%20y%20%5E%7B2%7D-%20l%20%2B2%20xi%20%7D%7B%20x%20%5E%7B2%7D%2B(%20y%20-%20l%20)%5E%7B2%7D%7D

x%5E%7B2%7D%2B(y-1)%5E%7B2%7D

%5Carg%20%5Cleft(%5Cfrac%7Bx%5E%7B2%7D%2By%5E%7B2%7D-1%2B2%20x%20i%7D%7Bx%5E%7B2%7D%2B(y-1)%5E%7B2%7D%7D%5Cright)%3D%5Ctan%20%5E%7B-1%7D%5Cleft(%5Cfrac%7B2%20x%7D%7Bx%5E%7B2%7D%2By%5E%7B2%7D-1%7D%5Cright)

%5Ctherefore%20%5Ctan%20%5E%7B-1%7D%5Cleft(%5Cfrac%7B2%20x%7D%7Bx%5E%7B2%7D%2By%5E%7B2%7D-1%7D%5Cright)%3D%5Cfrac%7B%5Cpi%7D%7B4%7D

%5CRightarrow%202%20x%3Dx%5E%7B2%7D%2By%5E%7B2%7D-1

%5CRightarrow%20(x-1)%5E%7B2%7D%2By%5E%7B2%7D%3D(%5Csqrt%7B2%7D)%5E%7B2%7D

%5Ctherefore

Locus of

%5Carg%20%5Cleft(%5Cfrac%7Bz%2Bi%7D%7Bz-i%7D%5Cright)%3D%5Cfrac%7B%5Cpi%7D%7B4%7D

is circle,
whose centre

(1%2C0)

and radius

%3D%5Csqrt%7B2%7D

Perimetre of circle

%3D2%20%5Cpi%20r%20%3D2%20%5Cpi(%5Csqrt%7B2%7D)

%3D2%20%5Cpi%20%5Csqrt%7B2%7D

Chapter Questions
3 MCQs