solve for z

metlx · #1 $Old$ November 5th, 2009, 02:31 PM

$z$ is not equal to $i$ (didn't find the "not equal to" sign)

Show that

$\mid \frac{i - z}{1 - i\overline{z}}\mid = 1$

I tried several times by putting in z = a + bi.. without luck. can someone show me how to do this?

tonio · #2 $Old$ November 5th, 2009, 02:49 PM

Quote:

Originally Posted by metlx $View Post$

$z$ is not equal to $i$ (didn't find the "not equal to" sign)

Show that

$\mid \frac{i - z}{1 - i\overline{z}}\mid = 1$

I tried several times by putting in z = a + bi.. without luck. can someone show me how to do this?

As you said is fine: $z=x+yi\Longrightarrow i-z=-x+(1-y)y\,,\,\,1-i\overline{z}=(1-y)-ix$

So you see the numerator's real parts equals the denominators imaginary part and the numerator's imaginary part equals the denominator's real part and thus the modulus of both denominator and denominator are equal...

Tonio

Drexel28 · #3 $Old$ November 5th, 2009, 02:57 PM

Quote:

Originally Posted by metlx $View Post$

$z$ is not equal to $i$ (didn't find the "not equal to" sign)

Show that

$\mid \frac{i - z}{1 - i\overline{z}}\mid = 1$

I tried several times by putting in z = a + bi.. without luck. can someone show me how to do this?

Problem: Compute $\left|\frac{i-z}{1-i\cdot\bar{z}}\right|$ given that $z\ne i$

Solution: Suppose that $z=a+bi$ then $\bar{z}=a-bi$ and $i\cdot\bar{z}=ai+b$ . So that $\left|\frac{i-z}{-i\cdot\bar{z}}\right|=\left|\frac{-a-(b-1)i}{(1-b)-ai}\right|=\frac{\left|-a-(b-1)i\right|}{\left|(1-b)-ai\right|}=\frac{\sqrt{a^2+\left(b-1\right)^2}}{\sqrt{a^2+\left(b-1\right)^2}}=1$

EDIT: Oops, sorry was a little late.