Year 12s: what will be the first way you research your future options? >>

Explanation Required

I'm having a little trouble understanding the concept behind the following past paper question (question and mark scheme shown below)

It's from FP2, June 2013 (R).

I can rearrange the equation and compare coefficients to get the u = 2/x and v+1 = 0; no problem with that. But I don't quite understand what this transformation looks like / represents or why the u component is ignored.

Sorry if I'm being a bit thick or missing something really obvious!

Cheers in advance.

EDIT: I may have stumbled upon it, I think. Is it basically that u varies with x, but that all points in the w plane always lie on the line v = -1 ?

(edited 9 years ago)

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Reply 1

theOldBean

A general point in the complex plane is given by

x+yi

. This is transformed to

\frac{x+yi+2i}{i(x+yi)}=\frac{x+(y+2)i}{ix-y}

. We are interested in the image of the real axis, so we are only interested in the case when y=0. Therefore we have that

x+0i

is transformed to

\frac{x+2i}{ix}=1/i+2/x =-i+\frac{2}{x}

.

What is the locus of

-i+2/x

as x varies? Notice that x is real, so this quantity ALWAYS has imaginary part -i. Now if x can be any real number, then

\frac{2}{x}

can be any real number except zero. So the image of the real axis in the w-plane is the horizontal line, one below the real axis, with the point

(0,-i)

removed.

Since the Q just asks for the equation of the line, its just

Im(z)=-1

which is equivalently written as v+1=0.
**EDIT: I actually mean

Im(w)=-1

since we are in the w-plane. Sorry!**

Does this help?

(edited 8 years ago)

Reply 2

Thevesh Theva

Perfect, theOldBean! Crystal clear and simple. Not only has your explanation helped me with this question, it's given me a framework to view other questions of this sort with.

Thank you for the help! :smile: