 Hello and welcome to the session. In this session we will discuss a question which says that a variable complex number z is such that amplitude of 2z plus 3 over 1z minus 1 is always equal to pi by 4. Illustrate the locus of z in argon plane. Now before starting the solution of this question, we should know our result. And that is an amplitude of the complex number z which is equal to a plus b iota is an angle theta with initial site the positive x axis and terminal site the way from the region containing a plus b iota. Thus is equal to amplitude of z which is equal to tan inverse modulus of b by a. Now this result will work out as a key idea for solving out this question. And now we will start with the solution. Now it is given that that is a variable complex number and also it is given the amplitude of this is always equal to pi by 4. Now let z is equal to x plus y iota then plus 3 whole upon z minus 1 will be equal to Now putting the value of z here it will be 2 into x plus y iota the whole plus 3 whole upon x plus y iota the whole minus 1 which is equal to 2x plus 3 the whole minus 1 the whole plus y iota. Now rationalizing this expression we get plus 3 the whole plus 2y iota whole upon x minus 1 the whole plus y iota. This expression we will multiply the numerator and denominator by the conjugate of this. So multiplying the numerator and denominator by x minus 1 the whole minus y iota over x minus 1 the whole minus y iota. Plus 3 the whole the whole into 1 the whole minus y iota the whole whole upon plus y iota the whole into x minus 1 the whole minus y iota the whole. So this will be equal to 2x plus 3 the whole into x minus 1 the whole plus 3 the whole into minus y iota plus 2y iota into x minus 1 the whole the whole plus into minus y iota. And here in the denominator we will use the formula of a plus b the whole into a minus b the whole which is equal to a square minus b square and here a is x minus 1 and b is y iota. So this will be equal to x minus 1 whole square minus y iota whole square. It will be equal to minus 2x minus 3y plus 2 minus y square iota square. Now this is equal to x minus 3 plus iota into now these terms are cancelled with each other. So it will be minus 5y here and here iota square is minus 1 and minus into minus is plus. So it will be plus 2y square square is equal to minus 1 whole upon and minus into minus is plus so it will be plus y square. So this is equal to 2x square plus 2y square plus x minus 3 minus 5y iota whole upon whole plus y square whole upon z minus 1 is equal to minus 3 the whole whole upon x minus 1 whole square plus y square minus whole upon x minus 1 whole square plus y square. Now it is given that it is always equal to pi by 4. Now using the result which is given in the key idea, compute 12 upon z minus 1 is equal to b by a, which is equal to theta upon z minus 1 is equal to pi by 4. So this implies modulus of b by a is equal to pi by 4. This b values here, this implies whole square plus y square minus 3 whole upon x minus 1 whole square plus y square modulus. This implies square plus x minus 3 is equal to plus x minus 3. Now cross contouring this implies pi by a is equal to 2x square plus 2y square plus 2y square minus pi by minus, now dividing throughout by 2 plus y square plus 1 by 2x minus pi by 2y. So the general equation of the circle is this, where the coordinates of center are minus g minus f, becomes g square plus. Now comparing this equation with the general equation of the circle is equal to minus pi by 2, which implies g is equal to 1 by 4, 5 by 4. Therefore, the coordinate is 1 by 4 minus 5 by 4 will be 5 by 4. This 0.25, 1 point of g square plus x square is equal to 5 root 2 by 4 units. So the radius equal to 1.76, which is equal to 1.8 units, 1.25 and radius 1.8 units approximately. Move the locus of z in urban plain. So this is the locus of z, which is a circle with center 0.25, 1.25, 0.8 units. And that's all for this session. Hope you all have enjoyed this session.