 Hello and welcome to the session. In this session we will learn about modulus of complex number and collinear points. First of all let us discuss modulus of complex number. Now the modulus or absolute value of a complex number z such that z is equal to a plus b iota where n b belong to the set of real numbers is given by modulus of z is equal to modulus of a plus b iota which is equal to square root of a square plus b square. Now we know that in the complex number z which is equal to a plus b iota a is called a real part of the complex number z and b is called the imaginary part of the complex number z. So modulus of z is equal to square root of the square of the real part of z the square of the imaginary part of z thus modulus of z is greater than equal to 0 that it belongs to the set of complex numbers and the modulus a plus b iota the distance from the origin to the point the modulus of a complex number z which is equal to a plus b iota diagrammatically. Now here the point is a b is representing the number that is the complex number z which is equal to a plus b iota. Now the distance over a that is the distance of the point a whose coordinates are a b from the origin is equal to the modulus of z which is equal to square root of a square plus b square. Let's discuss one example. Now here let the complex number z is equal to 4 plus 5 iota so modulus of z will be equal to square root of a square plus b square. Now here a which is the real part of z is equal to 4 and b which is the imaginary part of z is equal to 5. So modulus of z is equal to square root of a square that is 4 square plus b square that is 5 square. So this is equal to square root of 16 plus 25 which is equal to root 41. Now let us find the modulus of cos theta plus iota sin theta. So this is equal to square root of a square. Now a where is cos theta so it will be cos square theta plus b is sin theta so b square will be sin square theta. Now this is equal to now we know that cos square theta plus sin square theta is 1 so this is equal to root 1 which is equal to 1. Now let us discuss one theorem z to a hb is equal to modulus of z1 minus the complex number z1 is equal to plus q iota and the complex number sub 2 is equal to x plus y iota from the z1 which is equal to b plus q iota is represented by the point a whose coordinates are bq and the complex number z2 which is equal to x plus y iota is represented by the point b whose coordinates are now by distance formula is equal to square root of 2 is equal to b plus q iota the whole minus x plus y iota the whole which is equal to minus y the whole into iota. Now modulus of z1 minus z2 is equal to square root of 3 square now a here is p minus x and b here is y so modulus of z1 minus z2 will be equal to a square which is p minus y whole square. Now from this equation y whole square is equal to the distance a b therefore the distance a b is equal to modulus of z1 minus z2. Now three points are said to be collinear on the same straight line. So here we can say that the points p q equals p q plus the distance q i then the points p q and i are called collinear points. Now let us discuss one example in this we have to prove that the representative numbers start with its solution. Now let the complex number 2 plus a iota by the point p whose coordinates are 2 8 and the complex number 4 plus 14 iota is represented by the point q whose coordinates are is represented by the point r whose coordinates are that the points p q and r are collinear points that is this condition of collinearity and if this condition is satisfied then the three points are collinear of non collinear. So let us find out the distance now the distance p q is equal to square root of that is by the distance formula we can find out the distance p q which is equal to square root of the square of the difference of the abscissa that is the square of the difference of the coordinates which is 14 minus 8 whole square which is equal to square root of this 2 1 square will be 2 square and that is 4 whole square is 6 square that is and this is equal to root which can be written as now the distance q r the distance formula is equal to square root of this 14 whole square which is equal to square root of which is 36 this is equal to root 14 which is equal to 2 root 10. Now these are the coordinates of p and now the distance p r is equal to square root of 8 whole square which is equal to square root of which is equal to square root of 16 plus 1 44 which is equal to root 160 which is further equal to root 10. Now p q is equal to 2 root 10 q r is equal to 2 root 10 and p r is equal to 4 whole square root 10 is equal to 4 root 10 therefore q r is equal to the points p q and r polynomial points the clarity of so we have plotted the points p q and r on the graph the point p is representing the complex number 2 plus 8 iota the point q is representing the complex number 14 iota is representing the complex number by joining all the that means the points p q and r thus the points p q and r are called collinear points. So in this session we have learned about various of the complex number and collinear points.