Conics

Question 1

Circle

Answer 1

Plane is parallel to the base of the cones

Question 2

Ellipse

Answer 2

Plane is less steep than the surface of the cone

Question 3

Hyperbola

Answer 3

Plane is steeper than the surface of the cone so intersects both cones

Question 4

Parabola

Answer 4

Plane is parallel to the surface of the cone

Question 5

Eccentricity

Answer 5

How much a conic section varies from being circular

e = 0 for a circle

Question 6

Directrix

Answer 6

The line the curve would become if e = ∞

Question 7

Parabola definition

Answer 7

A locus of points equidistant from a line (directrix) and a point (focus)

Question 8

Horizontal parabola parametric and cartesian equations

Answer 8

Parametric: x = at^2, y = 2at
Cartesian: y^2 = 4ax

Question 9

Horizontal parabola focus, directrix, vertex

Answer 9

Focus: (a, 0)
Directrix: x = -a
Vertex: (0,0)

Question 10

Rectangular hyperbola

Answer 10

Curves with two perpendicular asymptotes, curve away as they move further from the origin
(x^2/a) + (y^2/a) = 1

Question 11

Rectangular hyperbola asymptotes x,y = 0 parametric and cartesian equations

Answer 11

Parametric: x = ct, y = c/t
Cartesian: xy = c^2
Looks like a reciprocal graph

Question 12

Ellipse parametric and cartesian equations

Answer 12

Cartesian: (x^2/a^2) + (y^2/b^2) = 1
Parametric x = a cosθ y = bsin θ

stretch of a circle by a parallel to x and b parallel to y

Question 13

Hyperbola cartesian and parametric equations

Answer 13

Cartesian: (x^2/a^2) - (y^2/b^2) = 1
Parametric: x = +/- a cosh t, y = b sinh t
x = a sec t, y = b tan t

Question 14

Hyperbola rules and facts

Answer 14

At y = 0, x = +/- a and x increases at an increasing rate as y increases
As x -> +/- ∞ (x^2/a^2) ≈ (y^2/b^2)
Asymptotes at y = +/- b/a x

Question 15

Eccentricity calculation and type

Answer 15

e = PS/PD (PS is distance from a point on the curve to the focus, PD to the directrix)
0 < e < 1 means ellipse
E > 1 means hyperbola

Question 16

Ellipse foci, directrices and eccentricity (if a > b)

Answer 16

For ellipse (x^2/a^2) + (y^2/b^2) = 1
Foci (+/- ae, 0)
Directrices: x = +/- (a/e)
b^2 = a^2(1-e^2)

Question 17

Ellipse/hyperbola if b>a

Answer 17

Rotate all 90°

Question 18

Hyperbola foci, directrices and eccentricity (if a > b)

Answer 18

For hyperbola (x^2/a^2) - (y^2/b^2) = 1
Foci (+/- ae, 0)
Directrices: x = +/- (a/e)
b^2 = a^2(e^2-1)