circles, ellipses & hyperbolas and exact value triangles for spec

diff --git a/spec/prelim.md b/spec/prelim.md
index ed06b09db8ad416577a72c0d21ee35889b9363de..c01fe95ec0f2b1a36c1eb4b857019f3092c7fc17 100644 (file)
--- a/spec/prelim.md
+++ b/spec/prelim.md
@@ -2,6 +2,9 @@
 
 ## Circular functions
 
 
 ## Circular functions
 

+![](../methods/graphics/exact-values-1.png){#id .class height=150px}
+![](../methods/graphics/exact-values-2.png){#id .class height=150px}
+

 $\sin \theta$ - $y$-coord on unit circle  
 $\cos \theta$ - $x$-coord on unit circle  
 $\tan \theta = {\sin \theta \over \cos \theta}$
 $\sin \theta$ - $y$-coord on unit circle  
 $\cos \theta$ - $x$-coord on unit circle  
 $\tan \theta = {\sin \theta \over \cos \theta}$


@@ -47,14 +50,31 @@ Sum of interior angles of $n$-sided polygon is $(n-2) \times 180^\circ$
 - ![](graphics/segment-angles.png){#id .class width=40%} angles in the same segment of a circle are equal
 - ![]()
 
 - ![](graphics/segment-angles.png){#id .class width=40%} angles in the same segment of a circle are equal
 - ![]()
 

+## Circles, ellipses and hyperbolas
+
+Standard form is $Ax^2+By^2+Cx+Dy=0$
+
+- if $A+B$ then circle
+- if $A>0$ and $B>0$ and $A\ne B$ then ellipse
+- if $A<0<B$ or $B<0<A$ then hyperbola

 
 

-## Ellipses and hyperbolas
+### Circles

 
 

-#### Ellipses
+$$(x-h)^2 + (y-k)^2 = r^2$$
+
+- centre $(h,k)$
+- radius $r$
+
+### Ellipses

 
 $${(x-h)^2 \over a^2}+{(y-k)^2 \over b^2} = 1$$
 
 
 $${(x-h)^2 \over a^2}+{(y-k)^2 \over b^2} = 1$$
 

-#### Hyperbolas
+- centre $(h, k)$
+- $x$-radius $a$
+- $y$-radius $b$
+- $\therefore \text{width}=2a, \quad \text{height}=2b$
+
+### Hyperbolas

 
 $${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$$
 
 
 $${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$$