Question

If $\left|\frac{z-25}{z-1}\right|=5$, then find the value of $\left|z\right|$

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is C

$5$

Explanation for the correct option.

Step 1. Simplify the given equation.

Let $z=x+iy$, then the equation $\left|\frac{z-25}{z-1}\right|=5$ becomes:

$$\begin{gathered} \left|\frac{z-25}{z-1}\right|=5 \\ \Rightarrow \frac{\left|x+iy-25\right|}{\left|x+iy-1\right|}=5\left[\left|\frac{a}{b}\right|=\frac{\left|a\right|}{\left|b\right|}\right] \\ \Rightarrow \left|\left(x-25\right)+iy\right|=5\left|\left(x-1\right)+iy\right| \\ \Rightarrow \sqrt{{\left(x-25\right)}^2+y^2}=5\sqrt{{\left(x-1\right)}^2+y^2} \end{gathered}$$

Now, square both sides and simplify.

$$\begin{gathered} {\left(\sqrt{{\left(x-25\right)}^2+y^2}\right)}^2={\left(5\sqrt{{\left(x-1\right)}^2+y^2}\right)}^2 \\ \Rightarrow x^2-50x+625+y^2=25\left(x^2-2x+1+y^2\right) \\ \Rightarrow x^2-50x+625+y^2-25x^2+50x-25-25y^2=0 \\ \Rightarrow -24\left(x^2+y^2\right)+600=0 \\ \Rightarrow -24\left(x^2+y^2\right)=-600 \\ \Rightarrow x^2+y^2=25 \end{gathered}$$

Step 2. Find the modulus of $z$.

For the complex number $z=x+iy$, the modulus is given as:

$\begin{array}{rcl}\left|z\right|& =& \sqrt{x^2+y^2}\\ & =& \sqrt{25}\left[x^2+y^2=25\right]\\ & =& 5\end{array}$

Thus the value of $\left|z\right|$ is $5$.

Hence, the correct option is C.