 Hello and welcome to the session. In this session we will discuss the following question that says, if alpha and beta are different complex numbers with modulus beta equal to 1, then find modulus of beta minus alpha upon 1 minus alpha bar into beta. Before we move on to the solution, let's discuss some results. First we have a complex number given by z into the conjugate of the complex number z that is z bar. This would be equal to modulus of z whole square. There the z is complex number. Now next we consider z1 z2 between complex numbers. Now we have z1 upon z2. This whole bar is equal to z1 bar upon z2 bar. We write it the complex number z2 is not equal to 0. We have z1 z2 this whole bar is equal to z1 bar plus minus z2 bar the idea of using this question. Let us now move on to the solution. In the question we have that alpha and beta are different complex numbers and modulus of beta is equal to 1, alpha and beta are two different complex numbers and we are also given that modulus of beta is equal to 1. We have to find modulus of beta minus alpha upon 1 minus alpha bar into beta. You have to find the value of this. We consider modulus of beta minus alpha upon 1 minus alpha bar into beta. The whole square can be key idea. We have that modulus of z square is equal to z into z bar which is the conjugate of the complex number. So using this result we can say that this is equal to beta minus alpha upon 1 minus alpha bar into beta and this whole into beta minus alpha upon 1 minus alpha bar into beta whole bar. So further this would be equal to beta minus alpha upon 1 minus alpha bar into beta this whole into conjugate of beta minus alpha upon 1 minus alpha bar into beta. Then z1 upon z2 the whole bar is equal to z1 bar upon z2 bar provided this complex number z2 is not equal to 0. So using this result beta minus alpha upon 1 minus alpha bar into beta and this whole bar and so this would be equal to beta minus alpha the whole bar upon 1 minus alpha bar into beta the whole bar. Other result which says that z1 plus minus z2 the whole bar is equal to z1 bar plus minus z2 bar this means that z1 minus z2 the whole bar is equal to z1 bar minus z2 bar. So using this result this is equal to beta bar minus alpha bar upon 1 bar beta the whole bar which is further equal to beta bar minus alpha bar minus alpha into beta bar. z1 upon 1 minus alpha into beta bar the whole that gives us beta minus alpha the whole into beta bar minus alpha bar the whole z1 upon into beta the whole into 1 minus into beta bar the whole. So further when we multiply these two expressions in the numerator we get beta into beta bar minus beta into alpha bar into beta bar plus alpha into alpha bar this whole upon when we multiply these two expressions in the denominator so we get 1 minus alpha into beta bar minus alpha bar into beta plus 2 alpha bar into beta into beta bar. So the modulus of the complex number whole square so this means beta into beta bar would be modulus of beta the whole square beta into alpha bar minus alpha into beta bar would be modulus of alpha the whole square. Now this whole upon into beta bar minus alpha bar into beta into alpha bar would be modulus of alpha the whole square and beta into beta bar would be modulus of beta the whole square. Now we are also given the modulus of beta is equal to 1 so using this we get this is equal to 1 square which is 1 minus beta into alpha bar minus alpha into beta bar plus modulus alpha the whole square this will upon into beta bar into beta modulus of alpha whole square is equal to 1 so modulus of beta whole square would be 1 so modulus of alpha whole square. We can observe that numerator and denominator are same so they both cancel with each other and we are left with 1 beta minus alpha alpha bar into beta whole square is equal to 1 this means that modulus of beta minus alpha upon 1 minus alpha bar into beta is equal to we were supposed to find the value of modulus of beta minus alpha upon 1 minus alpha bar into beta and this comes out to be equal to 1 so this is our final answer this can be instantiation hope you understood the solution of this question.