1.

Let `s , t , r`be non-zerocomplex numbers and `L`be the setof solutions `z=x+i y (x , y in RR, i=sqrt(-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?If `L`has exactlyone element, then `|s|!=|t|`(b) If `|s|=|t|`, then `L`hasinfinitely many elements(c) Thenumber of elements in `Lnn{z :|z-1+i|=5}`is at most2(d) If `L`has morethan one element, then `L`hasinfinitely many elementsA. If L has exactly one element , then `|s| ne |t|`B. If `|s|=|t|` then L has infinitely many elementsC. The number of elements in `L cap {z:|z-1+i|=5}` is at most 2D. If L has more than one element , then L has infinitely many elements

Answer» Correct Answer - A::C::D
We have
`sz+t barz+r=0`
On taking conjugate
`bars bar z+ bari z+ bar r`=0
On solving Eqs (i) and (ii) we get
`z=(barrt-rbars)/(|s^2-|t|^2`
(a) For unique solitions of z
`|s|^2-|t|^2 ne 0 rArr |s| ne |t|`
It is true
(b) If |s|=|t| then `bar r t - r bar s ` may or may not be zero so,z may have no soluiton L may be an empty set . It is false
( c) If elements ofset L repersents line then this line and given circle intersect at maximum two point Hence it is true .


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