 Hello and welcome to the session. In this session we will discuss a question which says that what are the representative points of the complex numbers 1-IOTA, 2-IOTA, 4-IOTA are collinear. Also draw the graph. Now before starting the solution of this question we should know some results. First is if the points p, q and r are collinear that is p, q and r are collinear then p2 plus qr is equal to pr. Second is if z1 and z2 are complex numbers and b on the ideal plane then the distance is given by ab which is equal to modus of z1 minus z2. Next number z which is equal to a plus b IOTA where a and b belong to set of real numbers. Modus of z is equal to modus of a plus b IOTA which is equal to square root of a square plus p square. Now these results will welcome you as a key idea. First solving out this question. We will start with the solution. Complex numbers are given to us. Now let the points p, q and r represent the complex numbers minus IOTA, 2 plus IOTA and 4 plus 5 IOTA respectively. Then by using this result which is given as a key idea. Now the distance p, q will be equal to modus of 1 minus IOTA the whole minus 2 plus IOTA the whole. This is represented by p and this is represented by q. Now this one is equal to modus of 1 minus IOTA minus 2 minus IOTA which is further equal to modus of minus 1 minus 2 IOTA. Now using this result which is given as a key idea minus 1 and b is equal to minus 2. So this will be equal to that is modus of minus 1 minus 2 IOTA will be equal to square root of a square plus b square which will be equal to minus 1 square plus minus 2 square which is equal to square root of 1 plus 4 which is equal to root 5. Now we will find the distance qr represented by q and this is represented by r. Now the distance qr will be equal to modus of IOTA the whole minus 5 IOTA the whole IOTA minus 4 minus 5 IOTA which is further equal to modus of minus 2 minus 4 IOTA square. Now a here is minus 2 and b here is minus 4 this will be equal to square root of 4 plus 16 which is fun. Now this number is represented by p will be equal to modus of minus 4 minus 5 IOTA which is further equal to modus of minus 3 minus 6. Now here a is equal to minus 3 and b is equal to minus 6. So this will be equal to which is minus 3 square plus b square which is minus 6 square root of 9 plus 36 which is equal to which is further equal to 3 root 5 is equal to 3 plus qr which is given in the key idea is equal to root pq which is equal to root 5 plus which is equal to 2 root 5 and this is equal to 3 root 5. Now pr is also 3 root 5 and pq plus qr is also 3 root 5. Therefore pq plus qr is equal to pr is equal to 3 root 5. The points pq is defined by the object here x, y of real address x and y. Now here in this IOTA represents x is 1 here and y is equal to minus 1. So this complex number is defined by the object here 1 minus 1 and represents the point q21. The complex number Now by drawing all the 3 points you can see that all the 3 points seem straight line. Therefore the given question and that's all for this session. Hope you all have a great day.