 Hi and welcome to the session. I am Arsha and I am going to help you with the following question which says if a plus i out of b is equal to x plus i out of whole square upon 2x square plus 1, prove that a square plus b square is equal to x square plus 1 whole square upon 2x square plus 1 whole square. Any two complex numbers z1 to the equal to a plus i out of b and z2 is equal to c plus i out of d are said to be equal, a is equal to c and b is equal to d that is the real part of z1 is equal to the real part of z2 and the imaginary part of z1 is equal to the imaginary part of z2. So, this idea we are going to use to solve this problem. So, this is a key idea. Let us now begin with the solution and here we have a plus i out of b is equal to x plus i out of whole square upon 2x square plus 1. Now, x plus i out of whole square is equal to x square plus 2x i out of plus i out of square upon 2x square plus 1. This is since a plus b whole square is equal to a square plus 2ab plus b square and here in case of a we have x and in case of b we have i. It can further be written as x square i out of square is minus 1 plus i out of square times 2x upon 2x square plus 1 and now separating the real and the imaginary parts we have x square minus 1 upon 2x square plus 1 plus i out of times 2x upon 2x square plus 1. Now comparing this with the standard form of the complex number which is a plus i out of b here we find that a is equal to x square minus 1 upon 2x square plus 1 and b is equal to 2x upon 2x square plus 1. Now let us find a square plus b square. So, a square will be x square minus 1 upon 2x square plus 1 whole square and b square is 2x upon 2x square plus 1 whole square. So, we are equal to x square minus 1 whole square upon 2x square plus 1 whole square plus 2x whole square upon 2x square plus 1 whole square. Now let us take lcm which is the common denominator 2x square plus 1 whole square and now opening the brackets in the numerator we have x raise to the power 4 minus 2x square plus 1 plus 4x square which is further equal to x raise to the power 4 plus 2x square plus 1 upon 2x square plus 1 whole square. This can further be written as x square plus 1 whole square upon 2x square plus 1 whole square and this is what we are required to prove that is a square plus b square is equal to x square plus 1 whole square upon 2x square plus 1 whole square. So this completes the solution. Hope you enjoyed it. Take care and have a good day.