If $\dfrac{z-12}{z-8i}=\dfrac{5}{3},$ set the value of the complex number $z.$

I started to multiply the complex fraction by the conjugate which I believe is $\dfrac{\bar z + 8i}{\bar z + 8i}.$

My problem is that the terms of the multiplication are getting hard, because I got thing like $|z|$.
Then I guess that the real part is going to be $\frac{5}{3}$ and the imaginary one zero.


  • 2
    $\begingroup$ Hint: $3(z-12) = 5(z-8i)$ $\endgroup$ – Hyperion May 31 at 0:22

You're overthinking a bit. Reduce this to standard algebra; solve for $z$. Recall your exercises with similar functions from middle school in terms of a real variable $x$.

I'll work out a similar problem to avoid giving you a direct answer to the given problem.

Find $z$ if $$\frac{z-2i}{z-3i} = \frac{5}{7}$$

We begin by "cross multiplying" - multiply both sides by the denominators of each side. Thus we get

$$\frac{z-2i}{z-3i} = \frac{5}{7} \implies 7(z-2i)=5(z-3i) \implies 7z-14i =5z-15i$$

Now we solve for $z$ by subtracting $5z$ from both sides, and adding $14i$ to both sides:

$$7z-14i =5z-15i \implies 2z = -i \implies z = - \frac 1 2 i$$

As a result, we get that

$$z = - \frac 1 2 i = 0 - \frac 1 2 i$$

whichever form you desire to use. A similar idea holds for your given problem


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.