Chapter 11: Analytic Geometry

When the plane is perpendicular to the axis of the cone when it intersects

When the plane is tilted slightly when it intersects the cone

When the plane is tilter farther so that it is parallel to the cone generator

When the plane intersects both parts (upper and lower nappes) of the cone

A collection of all points in a plane that are the same distance from a fixed point (focus) as they are from a fixed line (directrix)

Imaginary line that goes through the vertex and perpendicular to the directrix

(-{b/2a}, f(-{b/2a}))

A line that is ±2a units from the vertex

The value a, the directed distance from the vertex to the focus of the parabola

A line segment that passes through the focus and is parallel to the directrix, with the length of |4a|

1. Equation: (x-h)²=4a(y-k)

2. If a>0: opens up

3. If a<0: opens down

4. Vertex: (h, k)

5. Focus: (h, k±a)

6. Directrix: y=k±a

7. Axis of symmetry: x=h

1. Equation: (y-k)²=4a(x-h)

2. If a>0: opens right

3. If a<0: opens left

4. Vertex: (h, k)

5. Focus: (h±a, k)

6. Directrix: x=h±a

7. Axis of symmetry: y=k

A collection of all points in the plane whose distance from two fixed points is a constant sum

The point on the focal axis midway between the foci

Plural of focus, the fixed points equidistance form the center and located on the focal axis

The points where the ellipse intersects its axes

The line connecting vertices that passes through the center of an ellipse and through its two foci with the length of 2a

The line between the center of an ellipse that is perpendicular to the major axis

1. Equation: {x²/b²} + {y²/b²} = 1

2. Focal axis: x-axis

3. Foci: (±c, 0)

4. Vertices: (±a, 0)

5. Pythagorean relation: a²=b²+c²

1. Equation: {y²/a²} + {x²/b²} = 1

2. Focal axis: y-axis

3. Foci: (0, ±c)

4. Vertices: (0, ±a)

5. Pythagorean relation: a²=b²+c²

1. Equation: {(x-h)²/a²} + {(y-k)²/b²} = 1

2. Focal axis: y=k

3. Foci: (h±c, k)

4. Vertices: (h±a, k)

5. Pythagorean relation: a²=b²+c²

1. Equation: {(y-k)²/a²} + {(x-h)²/b²} = 1

2. Focal axis: x=h

3. Foci: (h, k±c)

4. Vertices: (h, k±a)

5. Pythagorean relation: a²=b²+c²

e=c/a

The collection of points P=(x, y) such that the distance of the distances from P to the foci is ±2a

d(F₁, P) - d(F₂, P) = ±2a

Fixed points of the hyperbola

The line through the foci

The point on the transverse axis midway between foci

The points where the hyperbola intersects the transverse axis

1. Equation: {x²/a²} - {y²/b²} = 1

2. Focal axis: x-axis

3. Foci: (±c, 0)

4. Vertices: (±a, 0)

5. Pythagorean relation: c²=a²+b²

6. Asymptotes: y = ±{b/a}x

1. Equation: {y²/a²} - {x²/b²} = 1

2. Focal axis: y-axis

3. Foci: (0, ±c)

4. Vertices: (0, ±a)

5. Pythagorean relation: c²=a²+b²

6. Asymptotes: y = ±{a/b}x

1. Equation: {(x-h)²/a²} - {(y-k)²/b²} = 1

2. Focal axis: y = h

3. Foci: (h±c, 0)

4. Vertices: (h±a, 0)

5. Pythagorean relation: c²=a²+b²

6. Asymptotes: (y-k) = ±{b/a}(x-h)

1. Equation: {(y-k)²/a²} - {(x-h)²/b²} = 1

2. Focal axis: x = h

3. Foci: (0, k±c)

4. Vertices: (0, k±a)

5. Pythagorean relation: c²=a²+b²

6. Asymptotes: (y-k) = ±{a/b}(x-h)

1. Ax² + Cy² + Dx + Ey + F = 0; A and C cannot both equal 0

2. Defines a parabola: AC = 0

3. Defines an ellipse: AC > 0

4. Defines a hyperbola: AC < 0

5. Defines a circle: A = C

Ax² + Bxy + Cy² + Dx + Ey + F = 0

1. x’ = rcos(α)

2. y’ = rsin(α)

cot(2θ) = {(A-C)/B}

If cot(2θ) ≥ 0: 0° < 2θ ≤ 90° and 0° < θ < 45°

If cot(2θ) < 0: 90° < 2θ < 180° and 45° < θ < 90°

1. Defines a parabola: B² - 4AC = 0

2. Defines an ellipse (or circle): B² - 4AC < 0

3. Defines a hyperbola: B² - 4AC > 0

For any point on a conic section, the ration between the distance from the focus and the distance to the directrix

e = {d(P₁, F)}/{d(P₂, D)}

1. Circle: e = 0

2. Parabola: e = 1

3. Ellipse: e < 1

4. Hyperbola: e > 1

e = c/a

r = {ep}/{1-ecos(θ)}

r = {ep}/{1+ecos(θ)}

r = {ep}/{1+esin(θ)}

r = {ep}/{1-esin(θ)}

Let x = ƒ(t) and y = g(t), where ƒ and g are two functions defined on some interval. Then a point on a graph (x, y) can be determined by finding x = ƒ(t) and y = g(t). Essentially, (x, y) = (ƒ(t), g(t)), where t is in the domain of ƒ and g

Successive values of t that show movement

1. Choose the easiest equation

2. Solve for the parameter

3. Substitute the result for the parameter of the other equation

4. The resultant equation is the entire graph

A projectile that is fired at an inclination θ to the horizontal, with an initial speed v₀, from a height h above horizontal

1. x(t) = (v₀cosθ)t

2. y(t) = -½gt²+(v₀sinθ)t+h

3. g = 9.8 m/sec² OR g = 32 ft/sec²