SAT Math Level 2 - COORDINATE GEOMETRY

Question 1

Ellipse

Answer 1

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

Question 2

General equation of an ellipse at origin

Answer 2

x intercepts at a and -a
y intercepts at b and -b
(x^2/a^2) + (y^2/b^2) = 1

Question 3

Equation of ellipse (changing the centre point)

Answer 3

(x-h)^2 / a^2 + (y-k)^2 / b^2 = 1
Centre point (h,k)

Question 4

General hyperbolas equation at origin

Answer 4

(x^2/a^2) - (y^2/b^2) = 1

Question 5

Equation of hyperbola (changing the centre point)

Answer 5

(x-h)^2 / a^2 - (y-k)^2 / b^2 = 1
Centre point (h,k)

Question 6

y-axis hyperbola

Answer 6

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

Question 7

Drawing hyperbolas

Answer 7

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

Question 8

3D coordinates on a plane in space

Answer 8

( x , y , z )

Question 9

3D distance between coordinates formula

Answer 9

d = square root (X2 - X1)^2 + (Y2 - Y1)^2 + (Z2 - Z1)^2

Question 10

Polar coordinates

Answer 10

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

Question 11

Polar coordinates with negative r values

Answer 11

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)