 Hello and welcome to the session. In this session, we will discuss a question which says that if z is equal to a plus v iota and alpha is equal to z plus z iota whole upon z minus z iota, then show that absolute value of alpha is equal to 1. Now before starting the solution of this question, we should know a result and that is the absolute value of modulus complex number z such that z is equal to a plus b iota where b belongs to the set of real numbers is given by the value of z is equal to absolute value of a plus b iota which is equal to square root of the modulus of this number is just the distance from the origin to this point. Now this result will work out as a key idea for solving out the solution. This given to us to let absolute value of alpha is equal to 1 equal to a plus b iota equal to z plus z iota whole upon z minus z iota. So we will find z plus z iota. Now putting the value of z here, this will be equal to a plus b iota plus a plus b iota the whole into equal to a plus b iota plus b iota square. This is equal to b iota plus a iota plus now iota square is equal to minus 1 so it has b into minus 1 because iota square is equal to minus 1. So further this will be equal to a plus b iota plus a iota minus b we will find z minus z iota. Now putting the value of z here this will be equal to a plus b iota the whole minus plus b iota the whole into which is further equal to a plus b iota minus a iota minus b iota square. Now this will be equal to a plus b iota minus z b into minus 1. So it will be minus 1 here will be equal to a plus b iota minus a iota plus b. And this can also be written as a plus b the whole plus further this can also be written as and from this taking minus 1 common it will be minus this b iota whole into iota. Now let us know this as 1 and this is z minus z iota putting the value of plus z iota we get i is equal to a plus b the whole plus a plus b the whole into iota whole upon a plus b the whole minus of a minus b the whole into iota. Now whole rational will be equal to into iota whole upon a plus b the whole is b the whole into iota. Now for rationalizing this we will multiply the numerator and denominator by the conjugate of this. So it will be into plus b the whole plus b the whole into iota whole upon a plus b the whole plus a minus b the whole into iota. So this implies alpha is equal to minus b the whole plus a plus b the whole into iota the whole into a plus b the whole plus a minus b the whole into iota the whole whole upon plus b the whole minus of a minus b the whole into iota the whole into a plus b the whole minus b the whole into iota the whole. Write the numerator so it will be a a-b the whole, into a-b the whole, plus, a-b the whole, into ta into ai太. Plus, plus lay the whole, into a-b the whole, into ai太. Plus, plus lay the whole, into ai太 into a-bthe whole, into ai太. Hold upon. Now the denominator, we will apply the formula of a-b the whole, into into A plus V the whole which is equal to A square minus V square. By applying the formula here this will be equal to A plus V whole square that is A square B square that will be into a delta whole square. Now this implies alpha is equal to A plus V the whole is equal to A square minus V square and here A minus V the whole into A minus V the whole will be A minus V whole square into iota plus A plus V the whole into A plus V the whole will be A plus V whole square into iota plus A plus V the whole into A minus V the whole will be equal to a square minus b square the whole into a lotta into a lotta will be a lotta square. Then whole square minus a whole square into a lotta square minus b square plus from these two terms taking a lotta common it will be b whole square plus a plus b whole square the whole into a lotta plus b square the whole into a lotta square is minus 1 b whole square minus b whole square which is equal to minus 1. Now this implies alpha is equal to minus b square plus here a minus b whole square will be a square plus b square minus 2 plus b square plus 2 a b the whole into a lotta and here it will be on multiple with minus 1 it will be plus b square minus a square whole square. Now in the denominator a plus b whole square is a square plus b square plus 2 a b into minus 1 this will be plus and here a square plus b square minus 2 a b. Now this implies now these terms are cancelled with each other and in the denominator it will be 2 a square plus 2 b square the whole into a lotta whole upon in the denominator these terms are cancelled with each other and it will be 2 a square plus 2 b square. Now this will be cancelled with each other so this implies alpha is equal to a lotta. Now we know that a complex number is a number of the form a plus b a lotta where a and b are real numbers and a is called the real part and b is called the imaginary part. This complex number is equal to 0 plus 1 a lotta equal to 0 and b is equal to 1 this result which is given in the t idea the absolute value of alpha is equal to square root of a square which is equal to square root of 0 square plus 1 square which is equal to root 1 and that is equal to 1 alpha is equal to 1. So this is the solution of the given question and that's all for this session hope you all have enjoyed the session.