Question

The radius of the circle is $\left|\frac{\left(z-i\right)}{\left(z+i\right)}\right|=5$ is given by

• A. $\frac{13}{12}$
• B. $\frac{5}{12}$
• C. $5$
• D. $625$

Answer 1

The correct option is B

$\frac{5}{12}$

Explanation for the correct option:

Calculating the radius of circle:-

Given Complex number-

$\left|\frac{\left(z-i\right)}{\left(z+i\right)}\right|=5$

Find-

Radius of circle $\frac{\left|z-i\right|}{\left|z+i\right|}=5$

Take $z=x+iy$

Now,

$$\begin{gathered} \left|x+\left(y-1\right)i\right|=5\left|x+\left(y+1\right)i\right| \\ {x}^2+{\left(y-1\right)}^2=25\left(x^2+{\left(y+1\right)}^2\right) \end{gathered}$$

On solving,

$$\begin{gathered} x^2+y^2-2y+1=25x^2+25y^2+50y+25 \\ 24x^2+24y^2+52y+24=0 \\ {x}^2+y^2+\left(\frac{13}{6}\right)y+1=0 \\ {\left(x-0\right)}^2+{\left(y+\frac{13}{12}\right)}^2=\left(\frac{169}{144}\right)-1 \end{gathered}$$

${\left(x-0\right)}^2+{\left(y+\frac{13}{12}\right)}^2=\frac{25}{144}$

This is of the form ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$.

On Comparing ;

$r=\frac{5}{12}$

Hence , The radius of the circle is $\left|\frac{\left(z-i\right)}{\left(z+i\right)}\right|=5$ is given by $\frac{5}{12}$.

Therefore,the correct answer is option (B)