ATAR Notes: Forum
VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: jane1234 on November 09, 2011, 07:58:48 pm
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Okay, just did exam 2 VCAA 2006...
Question 5 a ii
Write down the complex equation of the straight line which passes through the points z1 and -z1 in terms of conj(z1).
So this line has cartesian equation y=x
And conj(z1) = cis(-pi/4)
Answer is |z- conj(z1)| = |z + conj(z1)|
I have no idea how they derived that equation from the cartesian form other than trial and error. Someone please help ASAP! :D
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BUMP.
Sorry, could someone please explain this as the exam is tomorrow??
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If you give me a bit I will have a look, or if someone else is quicker then all the better :)
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Ok, are you familiar with questions that ask you to find the cartesian equation of things like:
|z-1| = |z-i|.... etc? Basically, if you have an expression in this form: |z-a| = |z-b|, where a and b are complex, then it represents a straight line that perpendicularly bisects the points a and b. You can see this because it is saying that, the distance from z to a is the same as the distance from z to b.
So we know that the straight line y=x, is the perpendicular bisector of the two points *z1 and -*z1, so we simply put this into our known expression to get the answer.
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Ok, are you familiar with questions that ask you to find the cartesian equation of things like:
|z-1| = |z-i|.... etc? Basically, if you have an expression in this form: |z-a| = |z-b|, where a and b are complex, then it represents a straight line that perpendicularly bisects the points a and b. You can see this because it is saying that, the distance from z to a is the same as the distance from z to b.
So we know that the straight line y=x, is the perpendicular bisector of the two points *z1 and -*z1, so we simply put this into our known expression to get the answer.
Yes I get it now... :) Thanks heaps!!
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even though dc302 already explained it here's my take on it.
so the line y=x is basically a perpendicular bisector of 2 points in the 4th and 2nd quadrant that and reflections of each other in the y and x axis.
the point conj(z1) is in 4th quad. -conj(z1) is in 2nd quad.
So
its simply
|z-conj(z1)|=|z+conj(z1)|
Just wondering, would this be accepted as an answer aswell:
Im(z)= Re(conj(z1)) x Re(z)
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Just wondering, would this be accepted as an answer aswell:
Im(z)= Re(conj(z1)) x Re(z)
I'm not sure what they would accept tbh, but it seems unlikely.