ellpises and hyperbolas for spec

diff --git a/spec/prelim.md b/spec/prelim.md
index d53e5ef2d6b261175a92fe0a18ac49e3d2ba18c0..586fc59e6d0ca5ad207c321cb3c20e159d9a2263 100644 (file)
--- a/spec/prelim.md
+++ b/spec/prelim.md
@@ -22,6 +22,10 @@ $$a^2=b^2 - 2bc \cos A$$
 
 ## Geometry
 
 
 ## Geometry
 

+### Area of a triangle
+
+$A={1 \over 2} a b \sin C$
+

 ### Parallel lines
 
 If parallel lines are crossed by transversal:
 ### Parallel lines
 
 If parallel lines are crossed by transversal:


@@ -38,7 +42,24 @@ Sum of interior angles of $n$-sided polygon is $(n-2) \times 180^\circ$
 
 ### Circle geometry
 
 
 ### Circle geometry
 

-- ![](graphics/circle-centre-angles.png) The angle at the centre of a circle is twice the angle at the circumference subtended by the arc
-- ![](graphics/semicircle-right-angle.png) the angle in a semicircle is a right angle
-- ![](graphics/segment-angles.png) angles in the same segment of a circle are equal
+- ![](graphics/circle-centre-angles.png){#id .class width=40%} The angle at the centre of a circle is twice the angle at the circumference subtended by the arc
+- ![](graphics/semicircle-right-angle.png){#id .class width=40%} the angle in a semicircle is a right angle
+- ![](graphics/segment-angles.png){#id .class width=40%} angles in the same segment of a circle are equal

 - ![]()
 - ![]()

+
+
+## Ellipses and hyperbolas
+
+#### Ellipses
+
+$${(x-h)^2 \over a^2}+{(y-k)^2 \over b^2} = 1$$
+
+#### Hyperbolas
+
+$${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$$
+
+- centre at $(h,k)$
+- asymptotes at $y-k=\pm{b \over a}(x-h)$
+
+${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$ and ${(y-k)^2 \over b^2} - {(x-h)^2 \over a^2} = 1$ are **conjugate hyperbolas**
+