Question

Assertion :Centre of circle $\left|\frac{z+1}{z-1}\right|=2$ is $\frac{5}{3}$ Reason: Radius of circle $\left|\frac{z-1}{z+1}\right|=2$ is $\frac{4}{3}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Let $z=x+iy$
Consider the equation
$|\frac{z+1}{z-1}|=2$
Or
$\frac{|z+1|}{|z-1|}=2$
Or
$|z+1|=2|z-1|$
$|z+1{|}^2+4|z-1{|}^2$
$(x+1{)}^2+(y{)}^2=4\left[\right(x-1{)}^2+y^2]$
$x^2+2x+1+y^2=4x^2-8x+4+4y^2$
$3x^2+3y^2-10x+3=0$
$x^2+y^2-\frac{10x}{3}+1=0$
$(x-\frac{5x}{3}{)}^2+y^2-\frac{25}{9}+1=0$
$(x-\frac{5}{3}{)}^2+y^2=\frac{16}{9}$
Or
$(x-\frac{5}{3}{)}^2+y^2=(\frac{4}{3}{)}^2$
Hence centre is at $z=\frac{5}{3}+0i$ and radius is $\frac{4}{3}$ units.
simillarly for the reason we will get radius = $4/3$