 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that show on argon diagram the complex number z is equal to minus of 2 plus 5 iota and its conjugate z star also illustrate z plus z star is real. Now let us start with the solution of this question. We know that on an argon plane the conjugate of a complex number is the reflection of the given number in x-axis. We will use this result to draw the conjugate of complex number that is equal to minus of 2 plus 5 iota. Let us first draw point p representing complex number z is equal to minus of 2 plus 5 iota on the plane. Now in this complex number we see that minus of 2 is the real part of the complex number and 5 iota is the imaginary part. So from region O we move 2 units left and 5 units up and we reach this point p with coordinates minus 2 5 and it represents complex number z is equal to minus of 2 plus 5 iota. To plot its conjugate point we will reflect this point in x-axis. Now reflection of point with coordinates xy is given by the point with coordinates x minus y in x-axis. So reflection of point p with coordinates minus 2 5 in x-axis is given by the point with coordinates minus 2 minus 5. So here we plot this point with coordinates minus 2 minus 5 and we name this point as point q and the coordinates of point q are minus 2 minus 5 which is the reflection of point p in x-axis and this point will represent the complex number minus 2 minus of 5 iota which will be equal to conjugate of complex number z and it is represented by z star thus conjugate of complex number z is equal to minus of 2 plus 5 iota is given by the complex number z star which is equal to minus 2 minus of 5 iota. Now we also need to show that z plus z star is real. First of all we shall join origin that is point o with point p as well as point q and we get o p and o q. Now we shall complete this parallelogram o p r q where o r is one of the diagonals of the parallelogram and we know that this point r represents the sum z plus z star and this point lies on x-axis which is the real axis thus point r is given by the coordinates minus 4 0 and point r represents complex number given by minus 4 plus 0 iota or simply minus 4 so we have z plus z star is equal to minus 4 and here we see that the imaginary part of z plus z star is 0 and z plus z star is equal to minus 4 which is a real number thus we can say that z plus z star is real which is the required answer this completes our session hope you enjoyed this session