Let barb z+b barz=c, b ≠ 0, be a line in the complex plane, where barb is the complex conjugate of b. If a point z1 is the reflection of a point z2 through the line, then barz1 b+ z2 barb=

Question

Let barb z+b barz=c, b ≠ 0, be a line in the complex plane, where barb is the complex conjugate of b. If a point z1 is the reflection of a point z2 through the line, then barz1 b+ z2 barb= (A) 4 c (B) 2 c (C) c (D) None of these

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<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/7d29a199627beabe1820950abb62a45f-.png" /> The given line is barb z+b barz=c (1) Let A(z1) be a reflection of B(z2) in the line (1). Let P(z) be any point on the line (1). We have, A P=B P ⇒ |A P|2=|B P|2 ⇒ |z-z1|2=|z-z2|2 ⇒ (z-z1)( barz- barz1)=(z-z2)( barz- barz2) ⇒ ( barz2- barz1) z+(z2-z1) barz+z1 barz-z2 barz2=0 (2) Since (1) and (2) represent the same line, we get ( barb/ barz2- barz1)=(b/z2-z1)=(c/z1 barz1-z2 barz2)=k( say ) ⇒ k( barz2- barz1)= barb, k(z2-z1)=b, k(z1 barz1-z2 barz2)=c Now, barz1 b+ z 2 barb = barz1 k(z2-z1) +z2 k( barz2- barz1) =k barz1 z2-z1 barz1+ barz2 z2-z2 barz1 =k(z2 barz2-z1 barz1)=c

Q. Let $\overset{ˉ}{b} z + b \overset{z}{ˉ} = c, b \neq = 0$ , be a line in the complex plane, where $\overset{ˉ}{b}$ is the complex conjugate of $b$ . If a point $z_{1}$ is the reflection of a point $z_{2}$ through the line, then $\overset{z}{ˉ}_{1} b +$ $z_{2} \overset{ˉ}{b} =$

$4 c$

$2 c$

$c$

None of these

Q. Let bˉz+bzˉ=c,b=0, be a line in the complex plane, where bˉ is the complex conjugate of b. If a point z1​ is the reflection of a point z2​ through the line, then zˉ1​b+ z2​bˉ=

4c

2c

c

None of these

Q. Let $\overset{ˉ}{b} z + b \overset{z}{ˉ} = c, b \neq = 0$ , be a line in the complex plane, where $\overset{ˉ}{b}$ is the complex conjugate of $b$ . If a point $z_{1}$ is the reflection of a point $z_{2}$ through the line, then $\overset{z}{ˉ}_{1} b +$ $z_{2} \overset{ˉ}{b} =$

$4 c$

$2 c$

$c$