# Conic Sections formulae using Complex Numbers

I have been preparing for the JEE Examination here in India and have been studying Complex Numbers for the past few days. One of the topics which falls under Complex Numbers is their application in Coordinate Geometry (Conic Sections). These include the following:

Equation of lines in different forms (parametric/non-parametric) Equations of Circles with various conditions (for example centered at $z_0$, or orthogonal to another circle and so on) Equations of Ellipses and Hyperbolas and so on.

 There are several other applications/equations mentioned which I haven't written here. The trouble I'm facing is that the book I use for Complex Numbers (Algebra for JEE Main and Advanced, by SK Goyal) only has the formulae listed without showing the derivations. I find it extremely hard to just accept and use a formula/result without knowing how it came into existence.



I would be grateful if somebody could please mention a source/reference book from which I could actually learn how to derive/reach all the formulae used for represinting Conics using Complex Numbers. The book does not necessarily have to match the level of the JEE Advanced Examination; it can be higher than that too. However I would prefer it if the book was at the level suitable for the JEE Advanced level only.

Many thanks in advance!  Edit: The results mentioned in my book are as follows: 

$1)$The equation of the line joining $z_1$ and $z_2$ is $$z(\bar{z_1}-\bar{z_2})-\bar{z}(z_1-z_2)+ z_1\bar z_2-z_2\bar z_1=0 \text{ (non parametric form).}$$



$2)$Three points are collinear if $$\begin{vmatrix}z_1&\bar{z_1}&1\\z_2&\bar {z_2}& 1\\z_3&\bar {z_3}& 1\end{vmatrix}=0$$



$3)$$\bar{a}z+\bar z a+b=0 where b\in \mathbb R describes the equation of a straight line (I don't know what a is, nor what \iota b is).$$$$4) The real and complex slope (I don't know what is meant by 'real' and 'complex') of the line \bar{a}z+\bar z a+b=0 are -\dfrac{\Re(a)}{\Im(a)} and -\dfrac{a}{\bar a} where b\in \mathbb R.$$$$5) If the lines \bar{a}z+\bar z a+k_1=0 and \bar{b}z+\bar b a+k_2=0 k_1,k_2\in \mathbb R are perpendicular to each other, then$$\bar{a}b+\bar ba=0$$6) z\bar z +a\bar z +\bar az+k=0 where k\in \mathbb R represents a circle with center -a and radius \sqrt{|a|^2-k}.$$$$7) If |z-z_1|+|z-z_2|=2a where 2a>|z_1-z_2| then z represents an ellipse with foci at z_1 \text{ and }z_2 and a\in \mathbb R^+ .$$$$8) If |z-z_1|-|z-z_2|=2a where 2a<|z_1-z_2| then z represents a hyperbola with foci at z_1 and z_2.$$$$9) Equation of all circles orthogonal to |z-z_1|=r_1\text{ and }|z-z_2|=r_2 is (nothing further is mentioned).$$$$10) \left |\dfrac{z-z_1}{z-z_2}\right | = k is a circle if k\neq 1 and will represent a line if k=1.$$$$11) The equation |z-z_1|^2+|z-z_2|^2=k will represent a circle if k\geq \frac12 |z_1-z_2|^2.$$$12)$If$\arg\left(\dfrac{(z_2-z_3)(z_1-z_4)}{(z_1-z_3)(z_2-z_4)}\right)=0, \pm \pi$, then the points$z_1,z_2,z_3,z_4\$ are concyclic.

• It might help if you would write out a few of the formulas, so we would know what sort of reference to look for. – Gerry Myerson Oct 7 '16 at 12:04
• You could also see whether math.stackexchange.com/questions/481582/… is any help, or math.stackexchange.com/questions/786215/…, or people.eecs.berkeley.edu/~wkahan/Math185/Conics.pdf, or anything else you get by typing Conic Sections formulae using Complex Numbers into the internet. – Gerry Myerson Oct 7 '16 at 12:07
• @GerryMyerson Thanks for responding Sir. I've edited my question to include the results mentioned in the book. These were actually mentioned as extra points, but these are frequently used to solved questions in the JEE papers. I would be truly grateful if you could please show me how to derive all these results, or give me a source from which I could learn how to derive them. Once again, many thanks Sir! – Ishan Oct 7 '16 at 14:01
• @GerryMyerson PS. Sir I've tried multiple times to format the edit properly, but it just isn't working. I hope you will excuse it Sir. – Ishan Oct 7 '16 at 14:07
• Have you looked at any of the links I gave? Were any of them helpful? – Gerry Myerson Oct 7 '16 at 21:24