If α and β are different complex numbers with |β| = 1, then find $$| \dfrac{\beta-\alpha}{1-\alpha\beta}|=1$$

Asked by Abhisek | 1 year ago |  142

##### Solution :-

Let α = a + ib & β = x + iy

It is given that, β = 1

$$\sqrt{x^2+y^2}=1$$

$$x^2+y^2=1$$

$$| \dfrac{(x+iy)-(a+ib)}{1-(a-ib)(x+iy)}|$$

$$| \dfrac{(x-a)-i(y-b)}{1-(ax+aiy-ibx+by)}|$$

$$| \dfrac{(x-a)-i(y-b)}{(1-ax-by)+i(bx-ay)}|$$

$$\dfrac{\sqrt{(x-a)^2+(y-b)^2}}{\sqrt{(1-ax-by)^2(bx-ay)^2}}$$

Hence, proved

Answered by Pragya Singh | 1 year ago

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