Slightly confused with the min/max thing in loci questions, what exactly does this mean? For example the question below....
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Complex Numbers Loci Question watch
- Thread Starter
- 05-04-2011 21:08
- 05-04-2011 22:42
For part a, you should get a circle that touches the y axis. This means that z could potentially be anywhere on the edge of that circle. For some value of z, arg(z) will be the angle between the positive real axis and a line connecting z with the origin.
If you consider different values of z on the circle then then each value will have an argument that is at least pi/2 but certainly smaller than pi. You'll notice that values of z with smaller arguments lie towards the right hand side of the circle and you get bigger arguments at the bottom-left of the circle.
But we need to make this more mathematically precise. Imagine you drew a straight line going from the origin outwards. Every point on that line would have the same argument. If you change the direction of that line anticlockwise then the argument increases, and when the line intersects your circle then the argument associated with your straight line is an argument that z could potentially have.
So you're interested in which gradients of that line intersect with the circle (because once you have a range of gradients, you can then get a range of arguments) Notice that at the maximum and minimum gradients of your line, the line actually touches the circle.
Let's suppose that, putting the circle into x and y coordinates, you get an equation for the circle in the form (x-a)^2 + (y-b)^2 = r^2 where you should be able to determine a, b and r. And let's suppose that your line has an equation y=kx where you don't yet know k.
If the line and the circle touch, then solving the simultaneous equations should give a repeated root. This means that if you take (x-a)^2 + (y-b)^2 = r^2 and replace each instance of y with kx (here we're using the substitution method) then you'll get a quadratic in x. If that quadratic has a repeated root then its discriminant will be zero. So what you want to do is find the values of k which make the discriminant zero.
Once you have the values of k, things will be easier. You can then sketch the lines on a graph then use trigonometry to find out the required angles.
This will mostly help you with part c. But if you are able to understand this then you should be able to get part b by looking at your sketch.