Let ð, ð, ð¥ and ð¦ be real numbers such that ð â ð = 1 and ð¦ â 0. If the complex number ð§ = ð¥ + ðð¦ satisfies Im((az+b/z+1))=y, then which of the following is(are) possible value(s) of ð¥?

Question

Let ð, ð, ð¥ and ð¦ be real numbers such that ð â ð = 1 and ð¦ â 0. If the complex number ð§ = ð¥ + ðð¦ satisfies Im((az+b/z+1))=y, then which of the following is(are) possible value(s) of ð¥? (A) -1+√1-y2 (B) -1-√1-y2 (C) 1+√1+y2 (D) 1-√1+y2

Tardigrade Admin · Accepted Answer

operatornameIm((a z+b/z+1))=y ⇒ operatornameIm((a(x+i y)+b/x+i y+1))=y ⇒ operatornameIm((a i y+(a x+b)/(x+1)+i y) × (((x+1)-i y)/((x+1)-i y)))=y ⇒ (a y(x+1)-y(a x+b)/(x+1)2+y2)=y ⇒ (a-b) y=y((x+1)2+y2) (x+1)2+y2=1 x=-1 ± √1-y2 (∵ a-b=1) x=-√1-y2

Q. Let $a, b, x$ and $y$ be real numbers such that $a - b = 1$ and $y \neq = 0$ . If the complex number $z = x + i y$ satisfies $I m (\frac{a z + b}{z + 1}) = y$ , then which of the following is(are) possible value(s) of 𝑥?

$- 1 + 1 - y^{2}$

$- 1 - 1 - y^{2}$

$1 + 1 + y^{2}$

$1 - 1 + y^{2}$

Q. Let a,b,x and y be real numbers such that a−b=1 and y=0. If the complex number z=x+iy satisfies Im(z+1az+b​)=y, then which of the following is(are) possible value(s) of 𝑥?

−1+1−y2​

−1−1−y2​

1+1+y2​

1−1+y2​

Im(z+1az+b​)=y ⇒Im(x+iy+1a(x+iy)+b​)=y ⇒Im((x+1)+iyaiy+(ax+b)​×((x+1)−iy)((x+1)−iy)​)=y ⇒(x+1)2+y2ay(x+1)−y(ax+b)​=y ⇒(a−b)y=y((x+1)2+y2) (x+1)2+y2=1 x=−1±1−y2​ (∵a−b=1) x=−1−y2​

Q. Let $a, b, x$ and $y$ be real numbers such that $a - b = 1$ and $y \neq = 0$ . If the complex number $z = x + i y$ satisfies $I m (\frac{a z + b}{z + 1}) = y$ , then which of the following is(are) possible value(s) of 𝑥?

$- 1 + 1 - y^{2}$

$- 1 - 1 - y^{2}$

$1 + 1 + y^{2}$

$1 - 1 + y^{2}$