HSC Stuff > HSC Mathematics Extension 2
Mathematics Extension 2 Challenge Marathon
wu345:
Let be a root of the equation , where
The root locus of a complex quadratic is the set of all points on the Argand Diagram that could be a root of the quadratic.
Sketch the root locus of
wu345:
Define .
i) Show that
ii) Show that
iii) Hence show that and find a similar expression for
iv) Hence show that
RuiAce:
--- Quote from: wu345 on April 06, 2017, 07:33:58 pm ---Let be a root of the equation , where
The root locus of a complex quadratic is the set of all points on the Argand Diagram that could be a root of the quadratic.
Sketch the root locus of
--- End quote ---
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Note that through these computations, the points (1,0) and (-1,0) are technically excluded. Coincidentally, the other case(s) brings it back.
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Note that the above line was an abuse of notation. Infinity is not a number.
Proof of the limit used on g(p)
(Brief) Explanation as to why the functions decrease/increase from -1 respectively
In a similar way, the other case of \( p\le -2\) shows that \( y=0\) is a part of the locus for all \( x > 0\).
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wu345:
Correct! Thanks for pointing out my flaw too, fixed it now. The intended shorter method however, was to simply sub the root into the quadratic and equate real and imaginary parts
RuiAce:
A classic complex numbers question I just did a few minutes ago.
HintYou only have to consider when \(|\alpha|=1\). After you prove it holds for \( |\alpha|=1\), the case \( |\alpha| < 1 \) falls out pretty easily from it
SpoilerYou need to know circle geometry as well as have a substantial understanding of complex number tricks.
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