JEE Advanced2018MathematicsComplex NumberActual
Let s , t , r be non-zero complex numbers and L be the set of solutions z = x + i y x , y ∈ ℝ , i = - 1 of the equation s z + t z - + r = 0 , where z - = x - i y . Then, which of the following statement(s) is (are) TRUE?
Options
- AIf L has exactly one element, then s ≠ t
- BIf s = t , then L has infinitely many elements
- CThe number of elements in L ∩ z : z - 1 + i = 5 is at most 2
- DIf L has more than one element, then L has infinitely many elements
Correct answer
A. If L has exactly one element, then s ≠ t
Step-by-step solution
Given s z + t z - + r = 0 ...(i) On taking conjugate s - z - + t - z + r - = 0 ...(ii) z s 2 - t 2 = r - t - r s - 1 If s ≠ t then z has unique value 2 If s = t then r - t - r s - may or may not be zero so L may be empty set 3 Locus of z is null set or singleton set or a line in all cases it will intersect given circle at most two points. 4 In this case locus of z is a line so L has infinite elements