You are Here: Home >< Maths

# FP2 Loci watch

1. 5. (a) Sketch on an Argand diagram the circle C whose equation is
| z - root3 - i | = 1
(b) Mark the point P on C at which |z| is a minimum. Find this minimum value.
(c) Mark the point Q on C at which arg z is a maximum. Find this maximum value.

I've done part (a) and drawn a circle with centre (root3, 1).
I don't know how to do part (b), surely this would be the the line from the origin to the circle with the shortest distance but how do I find it? Don't know how to do part (c) either.
2. |z| represents the distance from the origin. Your equation represents a circle. The max and min values of |z| are the distances from the origin of the points furthest/closest to the origin which are on that circle.

Once you've drawn the circle, start by drawing a line from the origin to the centre of the circle and then mark on the distances you know.
3. (Original post by thers)
5. (a) Sketch on an Argand diagram the circle C whose equation is
| z - root3 - i | = 1
(b) Mark the point P on C at which |z| is a minimum. Find this minimum value.
(c) Mark the point Q on C at which arg z is a maximum. Find this maximum value.

I've done part (a) and drawn a circle with centre (root3, 1).
I don't know how to do part (b), surely this would be the the line from the origin to the circle with the shortest distance but how do I find it? Don't know how to do part (c) either.
Finding the minimum would be by the use of Pythagoras' theorem, then the furthest distance would be shortest distance + diameter of circle.

Posted from TSR Mobile
4. Ok I got |z| to be 1 but in the back of the textbook where the answers are it shows the diagram but P is on the other side of the circle, furthest away from the origin and says |z| is 3.
5. (Original post by thers)
Ok I got |z| to be 1 but in the back of the textbook where the answers are it shows the diagram but P is on the other side of the circle, furthest away from the origin and says |z| is 3.
represents a circle with centre and a radius of 1.

= length of line from origin to centre of circle minus the radius.

= length of the line from origin to centre of circle add the radius.
6. (Original post by Joshmeid)
represents a circle with centre and a radius of 1.

= length of line from origin to centre of circle minus the radius.

= length of the line from origin to centre of circle add the radius.
so |z| 1 is correct?

So how do I do part (c) now?
I drew a line from the origin to the left of the circle. I created a right angled triangle with adjacent side root3 - 1 and opposite side 1 relative to the unknown angle. I get arctan(1/(root3 - 1)) but according to the textbook that is wrong - it should be pi/3.
7. b is correct.

For C, Look at your drawing, we can get the size of the angle between the positive x-axis and the |z| as we have the lengths for them, the maximum of argz will therefore be double that.
8. I don't understand. Where am I drawing a line from the origin to?
9. (Original post by thers)
I don't understand. Where am I drawing a line from the origin to?
Maybe an image of my own will help:

Now we can see triangle A is the same as triangle B hence the angles theta and the one above which is unnamed in my image are equivalent, hence we can work out what theta is and double it.
10. (Original post by Joshmeid)
Maybe an image of my own will help:

Now we can see triangle A is the same as triangle B hence the angles theta and the one above which is unnamed in my image are equivalent, hence we can work out what theta is and double it.
Got it now. ty

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: March 28, 2013
Today on TSR

### Three reasons you may feel demotivated right now

...and how to stay positive

### Can I get A*s if I start revising now?

Discussions on TSR

• Latest
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

## Groups associated with this forum:

View associated groups
Discussions on TSR

• Latest

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE