Least distance between two complex numbers on two loci

[moriheru]

Homework Statement

This is a CIE A'level maths P3 question out of an exam from 2013 in October/November. As there is no markscheme ( I at least can't find one), I would be grateful if someone could look at my solution to the problem and correct me if I made a mistake.
The problem is 8.(b) below.

Homework Equations

The Attempt at a Solution

The first locus they are asking for is that of a circle with centre (0,-1) and radius 1 and the second locus is a line 135 deg. to the horizontal (real number axis) starting at x=2. I call z₁ and z₂ the points which will give the least value of abs(z-w). Both these points must lie on a line l₂. My further working and sketch of loci and the line are in the following image.In the last step I use the distance formula for the two complex numbers I calculated in the earlier steps. In the earlier steps I equated the equation of l₂ and the equations for the loci.

Thanks for any effort! And sorry for the clumsy exposition!

 

[haruspex]

Homework Statement

This is a CIE A'level maths P3 question out of an exam from 2013 in October/November. As there is no markscheme ( I at least can't find one), I would be grateful if someone could look at my solution to the problem and correct me if I made a mistake.
The problem is 8.(b) below.
View attachment 195845

Homework Equations

The Attempt at a Solution

The first locus they are asking for is that of a circle with centre (0,-1) and radius 1 and the second locus is a line 135 deg. to the horizontal (real number axis) starting at x=2. I call z₁ and z₂ the points which will give the least value of abs(z-w). Both these points must lie on a line l₂. My further working and sketch of loci and the line are in the following image.In the last step I use the distance formula for the two complex numbers I calculated in the earlier steps. In the earlier steps I equated the equation of l₂ and the equations for the loci.
View attachment 195848

Thanks for any effort! And sorry for the clumsy exposition!

I get the same result. You can simplify it to √2 + 1/√2 - 1

 

[moriheru]

Thanks! That is great! Is my method correct and is there any general method how to approach these problems geometrically or is it just case by case observation? I know of a calculus based approach. As far as I know one expresses abs(z-w) in terms the general coordinates of any point on the loci and then sets the derivative equal to zero. Is this correct? And how would the general form for abs(z-w) look like? Thanks for any further effort?

 

[haruspex]

is there any general method how to approach these problems geometrically or is it just case by case observation?

A general method (not just circles and lines) will necessarily be by calculus. Of course, minimising |z-w| is the same as minimising |z-w|², which simplifies things a little.
In many cases, it will be a bit easier with a geometric approach. In this one, I did it by rotating the circle's centre through 45 degrees about the origin. Then I only needed the horizontal distance from the circle to the line x=√2.

 

[moriheru]

Thanks a lot ! And interesting approach. I will keep that in mind.