Let a , b , x and y be real numbers such that a - b = 1 and y 0 . If the complex number z = x + i y satisfies Im ( a z + b z + 1 ) = y , then which of the following is(are) possible value(s) of x ? [2017]

Question

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 $Im (\frac{a z + b}{z + 1}) = y,$ then which of the following is(are) possible value(s) of $x$ ? [2017]

Smart Achievers · Accepted Answer

(1, 2)

$a - b = 1, y \neq 0$

$Im (\frac{a z + b}{z + 1}) = y$

$\Rightarrow Im [(\frac{a (x + i y) + b}{(x + 1) + i y}) \cdot \frac{(x + 1) - i y}{(x + 1) - i y}] = y$

$\Rightarrow \frac{- (a x + b) y + a y (x + 1)}{{(x + 1)}^{2} + y^{2}} = y$

$\Rightarrow \frac{- a x y - b y + a x y + a y}{{(x + 1)}^{2} + y^{2}} = y$

$\Rightarrow a - b = {(x + 1)}^{2} + y^{2}$

$\Rightarrow 1 = {(x + 1)}^{2} + y^{2}$

$∴$ $x = - 1 \pm \sqrt{1 - y^{2}}$

Solution

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 $Im (\frac{a z + b}{z + 1}) = y,$ then which of the following is(are) possible value(s) of $x$ [2017]

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

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

3 $1 + \sqrt{1 + y^{2}}$

4 $1 - \sqrt{1 + y^{2}}$

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Solution

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(az+bz+1)=y, then which of the following is(are) possible value(s) of x [2017]

1 -1+1-y2

2 -1-1-y2

3 1+1+y2

4 1-1+y2