JEE MAIN - Mathematics (2019 - 12th April Evening Slot - No. 10)
Let z $$ \in $$ C with Im(z) = 10 and it satisfies $${{2z - n} \over {2z + n}}$$ = 2i - 1 for some natural number n. Then :
n = 20 and Re(z) = –10
n = 40 and Re(z) = 10
n = 40 and Re(z) = –10
n = 20 and Re(z) = 10
Explanation
Let Re (z) = x, then
$${{2(x + 10i) - n} \over {2(x + 10i) + n}} = 2i - 1$$
$$ \Rightarrow \left( {2x - n} \right) + 20i = - \left( {2x + n} \right) - 40 - 20i + 2ni$$
$$ \Rightarrow 2x - n = 2x - n - 40$$
& 20 = -20 + 2n $$ \Rightarrow $$ x = -10 & n = 20
$${{2(x + 10i) - n} \over {2(x + 10i) + n}} = 2i - 1$$
$$ \Rightarrow \left( {2x - n} \right) + 20i = - \left( {2x + n} \right) - 40 - 20i + 2ni$$
$$ \Rightarrow 2x - n = 2x - n - 40$$
& 20 = -20 + 2n $$ \Rightarrow $$ x = -10 & n = 20
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