SATHEE: Complex Numbers 1 Question 10

Complex Numbers 1 Question 10

10. 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 Adv.)

(a) $1 - \sqrt{1 + y^{2}}$

(b) $- 1 - \sqrt{1 - y^{2}}$

(d) $- 1 + \sqrt{1 - y^{2}}$

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Answer:

Correct Answer: 10. (b, d)

Solution:

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

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

$\begin{aligned} \Rightarrow \frac{(a - b) y}{(x + 1)^{2} + y^{2}} = y \\ ∵ a - b = 1 \\ ∴ (x + 1)^{2} + y^{2} = 1 \\ ∴ x = - 1 \pm \sqrt{1 - y^{2}} \end{aligned}$

Complex Numbers 1 Question 10

10. 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 $x$ ?

Answer:

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Complex Numbers 1 Question 10

10. 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 Imaz+bz+1=y, then which of the following is(are) possible value(s) of x ?

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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10. 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 $x$ ?