17 Conic Sections Flashcards
What do Parabols, Hyperbolas, Circles and Ellipses have in common?
They are all formed by quadratic equations.
How is a parabola formed?
By a directrix, a line in the plane, and a focus, a point not on the directrix.
A parabola is the set of all points equidistant from the directrix and the focus.
What 2 types of parabolas are there?
x² parabolas (face up or down) and
y² parabolas (face right or left - not functions).
What is a hyperbola?
A hyperbola has 2 pieces that are mirror images. It is formed by the intersection of a plane with a double cone.
When do hyperbolas arise in nature/observation?
(1) the curve y(x) = 1/x
(2) the path of a spacecraft using gravity-assisted swing-by of a planet
(3) the path of a comet travelling to fast to ever return to the solar system
Describe the shape of a hyperbola.
Each branch has 2 arms which become straighter further out from the center.
What does this create?
2 diagonally-opposite arms which serve as asymptotes.
Where they intersect is the center of symmetry for the whole hyperbola.
With y = 1/x, the asymptotes are the x- and y- axes.
What are the components of a Hyperbola?
It has 2 foci, a center, 2 asymptotes, semi-major axis, linear eccentricity and eccentricity
How is a Hyperbola formed?
It is the locus of points such that any point, P, such that the absolute value of the difference of the distance from the 2 foci is constant.
That constant is 2 times the semi-major axis or the distnce between the vertex of each branch.
What are the axis of symmetry, the vertices, the constant and the asymptotes of a Hyperbola?
The axis of symmetry is the line that goes through the 2 foci.
The vertices are where the 2 curves make their sharpest turn.
The constant difference is the distance between the 2 vertices.
The asymptotes are not part of the Hyperbola but show where the curve would go in infinity.
What is the equation for the asymptotes of a Hyperbola?
If it is centered at the origin, one vertex is a t (a,0) and the other is at (-a,0)
The 2 asymptotes are:
y = (b/a)x and
y = -(b/a)x
where b is the y-coordinate of any point on the branches and a is the x-coordinate.
Therefore, what is the equation for the Hyperbola?
x²/a² - y²/b² = 1
How does this vary from the equation for an ellipse?
ellipse: x²/a² + y²/b² = 1
* Think of an ellipse as an imploded hyperbola.*
How do you measure the eccentricity of a Hyperbola?
Eccentricity measures the ratio of the difference from the focus to the distance to the directrix
What is the equation for eccentricity?
e = √(a² + b²)/a
where a is the x-coordinate of any point and b is its y-coordinate.
How does eccentricity work at the vertex?
1 /1 + 0/1 = 1
What is the Latus Rectum of a Hyperbola?
The line which goes through the focus parallel to the Directrix.
Its length is 2b²/a
(= 2*y-coordinate/x-coordinate)
Is a Hyperbola a function?
No. It does not pass the Vertical Line test.
Describe the parameters of the Hyperbola:
y²/25 - x²/36 = 1
The branches are top and bottom, not left and right.
a = 5; b= 6
e = √(a² + b²)/a = 1.56
What are the vertices and asymptotes of:
81x² - 9y² = 729
81x² - 9y² = 729
81x²/729 - 9y²/729 = 1
x²/9 - y²/81 = 1
a = 3, b = 9
asymptotes = ±9/3x
vertices = ±3,0
Is this a parabola or a hyperbola:
x² + 5y = 7x
Parabola.
There is no y² term.
Is this a parabola or a hyperbola:
3x² - 4y² = 60
Hyperbola.
3x²/60 - 4y²/60 = 1
Is this a parabola or a hyperbola:
2y² = 4x² + 100
Hyperbola.
-4x² + 2y² = 100
y²/50 - x²/25 = 1
Does this hyperbola open along the x or y axis?
x²/16 - y²/49 = 1
It opens along the x-axis.
Its vertices and (4,0) and (-4,0)
center = origin
Does this hyperbola open along the x or y axis?
y²/9 - x²/64 = 1
the y-axis
the vertices are (0,3) and (0,-3)
center = origin
Does this hyperbola open along the x or y axis?
y²/81 - x²/25 = 1
y-axis
vertices = (0,9) and (0,-9)
center = origin
Describe the graph of x²/121 - y²/81 = 1
It opens along the x-axis.
Its vertices are (11,0) and (-11,0)
Its center is the origin
with a = 11 and b = 9 it is a fairly “square” hyperbola.
Describe the graph of:
x²/16 - y²/49 = 1
opens on the x-axis
vertices = (4,0) and (-4,0)
centers on origin
Describe the graph of:
y² - x²/16 = 1
it opens along the y-axis
y²/1 - x²/16 = 1
vertices (0,1) and (0,-1)
This is a flat hyperbola.
Describe the graph of:
y²/64 - x²/16 = 1
it opens along the y-axis
the vertices are (0,8) and (0,-8)
This is a tight parabola (?)
