Ellipse
Basically a squashed circle
- so there isn’t a constant radius or diameter
- the long and short diameters (major and minor axes) are all we need to get its equation
General equation of an ellipse at origin
x intercepts at a and -a
y intercepts at b and -b
(x^2/a^2) + (y^2/b^2) = 1
Equation of ellipse (changing the centre point)
(x-h)^2 / a^2 + (y-k)^2 / b^2 = 1 Centre point (h,k)
General hyperbolas equation at origin
(x^2/a^2) - (y^2/b^2) = 1
Equation of hyperbola (changing the centre point)
(x-h)^2 / a^2 - (y-k)^2 / b^2 = 1 Centre point (h,k)
y-axis hyperbola
If you have (y^2/b^2) - (x^2/a^2) = 1, (so the y is before the x); then the parabola will open up-down not left-right
Drawing hyperbolas
1) draw a box with centre (h,k), width 2a and height 2b
2) Draw 2 lines through the corners and centre of the box. These are the sloping asymptotes. The hyperbola will get closer to these but never cross them
3) Plot the two vertices to the left and right of the centre on the sides of the box
4) Draw the two branches of the hyperbola getting closer to the asymptotes
3D coordinates on a plane in space
( x , y , z )
3D distance between coordinates formula
d = square root (X2 - X1)^2 + (Y2 - Y1)^2 + (Z2 - Z1)^2
Polar coordinates
Polar coordinates are another way pf indicating the position of a point in the plane. Instead of the origin, we talk about the pole. The positive x-axis is now called the initial line
- The polar coordinates of a point are its distance r from the pole and its angle, o, a counter-clockwise rotation about the pole from the initial line.
- We write these as a pair ( r , o )
- polar coordinates for a given point are not unique, e.g. (4,30) and (4,390) alll represent the same point
Polar coordinates with negative r values
We define the polar coordinates of a point with a negative value of r to be the same as the positive value of r, but we add 180 degrees to the angle.
e.g. (3,45) = (-3,-135)
