Complex Numbers (1 Viewer)

[kaz1]

 

[shaon0]

Eauate imaginary and reals then make t the subject in the x equation. sub it into y equ. x=1+3t, y=2-4t. Draw a perpendicular bisector to the line from O.

 

[GUSSSSSSSSSSSSS]

let z = x + iy </br> therefore: x + iy = 1 + 2i + t(3 - 4i) </br> then compare real and imaginary parts to get: </br> x = 1 + 3t ................. y = 2 - 4t </br> and im sure u can solve those parametric equations xD </br> </br> </br> </br> to find minimum modulus just use the perpendicular distance formula for THAT locus and the origin xD

 

[kaz1]

lol, thx people, bit of a mental blank. Been cramming 4unit all day.

 

[youngminii]

Since t is real, z = (1 + 3t) + (2 - 4t)i
Equating real and imaginary parts (z = x + iy)
1 + 3t = x
1/3(x - 1) = t_______(1)
2 - 4t = y__________(2)

Substituting 1 into 2
2 - 4/3(x - 1) = y
6 - 4x + 4 = 3y
4x + 3y = 10

The modulus is the distance of a point from the origin.
This point can be found by using the perpendicular distance formula, giving the minimum distance.
d = abs[(0 + 0 + 10)/sqrt(4^2 + 3^2)]
= 10/5
= 2

Edit: Fixed up, thanks eldore. Didn't see the mistake (typing maths is a horrible process heh)

 

[youngminii]

I took way too long to do that question.. For only 4 marks

Speaking of modulus, how do you find the minimum/maximum modulus to a circle? I never found this one out lol

 

[eldore44]

I just solved for y when x = 0. and then x when y = 0. and added the two. kinda seems silly now that i think about it. but it worked...

 

[GUSSSSSSSSSSSSS]

I took way too long to do that question.. For only 4 marks

Speaking of modulus, how do you find the minimum/maximum modulus to a circle? I never found this one out lol

were you perhaps thinking minimum/maximum ARGUMENT to a circle???? that's more common...

 

[youngminii]

My bad, yes I meant argument.

Hey gus, I found out gurmies was white today
Apparently my reaction was the same as yours when you found out lol

 

[eldore44]

as for minimum modulus, 2/5 seems too small. I see it as closer to 2.

The modulus is the distance of a point from the origin.
This point can be found by using the perpendicular distance formula, giving the minimum distance.
d = abs[(0 + 0 + 10)/(4^2 + 3^2)]
= 10/25
= 2/5

I hope that last bit's correct..

ahh, perpendicular dist formula is over the SQROOT(4^2 + 3^2) hence =2

 

[kwabon]

My bad, yes I meant argument.

Hey gus, I found out gurmies was white today
Apparently my reaction was the same as yours when you found out lol

use tangents to the circle that go through the origin, and then work out the acute angle the x axis and the tangent. EDIT : WTF???!!! gurmies is white?????? nahhh kiddin, dont even know who the bloke is, i think i may have saw you on fb commenting away on namu status, hahaha

 

[biopia]

Here is an example of one of those circle questions as well as working =]
Draw a sketch of the region |z-√2-√2i|≤1 and hence find:
a) The maximum and minimum values of z
b) The maximum and minimum values of argz

The ≤1 bit means the circle will be solid, and it will be shaded on the inside.

Obviously the graph has its centre at (√2, √2) and has a radius of 1.
If you imagine a right angled triangle, it would have a hypotenuse of 2 units, as well as be isosceles.
This means the max/min values of z, will be 2-1 (as it is closest to the origin) = 1 and 2+1 (as it furthest from the origin) = 3. In case you didn't get that, it was just adding and subtracting the radius from the hypotenuse of the triangle that can be constructed.

For b), we need to construct two right angled triangles with tangents connecting the origin to the circle. These tangents represent the min/max argz.
Since a radius is at right angles to the tangents, and the hypotenuse has a length of two, you can use simple trig to calculate the small angle near the origin. You should get π/6. Add and subtract this to π/4 and you'll get the max/min values of argz to be π/12 and 5π/12.

I love these kind of questions, lol.
Hope it helps.

 

[GUSSSSSSSSSSSSS]

My bad, yes I meant argument.

Hey gus, I found out gurmies was white today
Apparently my reaction was the same as yours when you found out lol

yea those type of questions i cant really explain without a diagram, however biopia above has done a fair good job xD xD

HAHAHAHAHAHAHHAHAHAHAHA YEA I KNOW AYE?!?!??!! i was like WTF GURMIES IS WHITE!!?!?!?!?!?!?!?! i remember when i told namu he ALSO had a similar reaction!!!! lolllllll

 

[gurmies]

LOL xD