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Mathematics: Post your doubts here!

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darks

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Help please :) thanks
 
Nazirah Ahmad

Nazirah Ahmad

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Hello, any kind soul can help me with MJ/13/33/ Question 7 (ii) (the highlighted part) ?

I managed to get the equation before it, but don't know how to show the result from that equation, please help. Thanks ! :)

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qwertypoiu

qwertypoiu

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Nazirah Ahmad said:
Hello, any kind soul can help me with MJ/13/33/ Question 7 (ii) (the highlighted part) ?

I managed to get the equation before it, but don't know how to show the result from that equation, please help. Thanks ! :)

View attachment 59894
Click to expand...
Let's say you want to expand:
|z-a|^2
where a is a real number. Your expansion will be:
|z-a|^2 = (z-a).(z-a)* = (z-a).(z*-a*)
Note that since a is real, a* = a. So:
(z-a).(z*-a*) = (z-a).(z*-a) = zz* - az - az* + a^2.

For example:
|z-5|^2 = zz* - 5z - 5z* + 25

Notice that the coefficients of z term and z* term are both -5, and then the constant term is (-5)^2. (Also coefficient of zz* is 1)

Now let's say you wanted to expand:
|z - si|^2,
where si is an imaginary number. Your expansion will be:
|z-si|^2 = (z-si).(z-si)* = (z-si).(z*-(si)*)
Notice that (si)* = -si, (eg (5i)* = -5i). So:
(z-si).(z*-(si)*) = (z-si).(z*+si) = zz* + si.z - si.z* + s^2

For example:
|z-5i|^2 = zz* + 5iz - 5iz* + 25

Notice that the coefficient of z and z* is +5 and -5 respectively. They alternate in sign. Also, the constant term is +s^2 (where s is coefficient of the imaginary term)

So anyhow with enough practice you can learn the above formats and reverse it as if completing the square. So in summary:
|z-a|^2 = zz* - az - az* + a^2
|z - si|^2 = zz* + si.z - si.z* + s^2

So to your question. You have:
zz* - 2iz* + 2iz - 12 = 0 (Note that they have listed the z* term before z term. In my expansion above I did opposite.)
So this matches with:
|z - si|^2 = zz* + si.z - si.z* (+s^2)

We need to provide the s^2 term by adding 2^2=4 and then subtracting it. (just like when completing square):
zz* - 2iz* + 2iz - 12 = 0
zz* - 2iz* + 2iz +4 - 4 - 12 = 0
|z - 2i|^2 - 16 = 0
|z - 2i|^2 = 16
|z - 2i| = 4
 
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