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Level 499

Conics


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The definition of a conic
Graphs that are created by slicing a cone
4
The number of conics
Ancient Greeks
Who invented the conics
Parabola, circle, ellipse, hyperbola
The names of the conics
x^2 = 4py
The equation of a vertical parabola with vertex at (0,0)
y^2 = 4px
The equation of a horizontal parabola with vertex at (0,0)
What "p" represents in a parabola
The distance from the vertex to the focus AND the distance from the vertex to the directrix
The definition of a parabola
The set of points equidistant from a fixed point (the focus) and a fixed line (the directrix)
down
If a < 0, the parabola opens
Right
a positive exponent means the original decimal needs to move to the _______________________
(0, 4)
The focus in this parabola: x^2 = 16y
x = 1
The directrix in this parabola: y^2 = -4x
p = 3/2
The value of "p" in this parabola: x^2 = -6x
The definition of a circle
The set of points equidistant from a fixed point
x^2 + y^2 = r^2
The equation of a circle centered at (0,0)
radius = 9
The radius of this circle: x^2 + y^2 = 81
A fact we know about tangent lines and radii
A tangent line is always perpendicular to the radius at the point of tangency
slope of radius = 5/4
The slope of the tangent line to x^2 + y^2 = 25 at (4, 5)
The definition of an ellipse
The set of points such that the sum of the focal radii is a constant.
x^2/b^2 + y^2/a^2 = 1
The equation for a vertical ellipse centered at (0,0)
x^2/a^2 + y^2/b^2 = 1
The equation for a horizontal ellipse centered at (0,0)
The distance from center to vertex
The meaning of "a" in an ellipse
The meaning of "b" in an ellipse
The distance from center to endpoint (covertices)
The distance from center to focus
The meaning of "c" in an ellipse
a^2 - b^2 = c^2
The formula relating a, b, and c in an ellipse
How you decide whether an ellipse is vertical or horizontal looking at its equation
Find the largest denominator (which is a^2). If it is under the x, it's horizontal. If it's under the y, it's vertical.
Vertices are (0, 6) and (0, -6)
The vertices of x^2/9 + y^2/36 = 1
Endpoints are (0, 2) and (0, -2)
Endpoints (covertices) of x^2/25 + y^2/4 = 1
c = square root of 33
The c value for x^2/16 + y^2/49 = 1
The definition of a hyperbola
The set of points such that the absolute value of the difference of the focal radii is a constant.
y^2/a^2 - x^2/b^2 = 1
The equation of a vertical hyperbola centered at (0, 0)
x^2/a^2 - y^2/b^2 = 1
The equation of a horizontal hyperbola centered at (0, 0)
The way to determine whether a hyperbola is vertical or horizontal by looking at its equation
Look at what variable is first (positive). If x is first, it's horizontal. If y is first, it's vertical.
The distance from center to vertex
The meaning of "a" in a hyperbola
The distance from center to endpoints
The meaning of "b" in a hyperbola
The distance from center to focus
The meaning of "c" in a hyperbola
a^2 + b^2 = c^2
The equation relating a, b, and c in a hyperbola
m = (plus or minus) a/b
The formula for the slopes of the asymptotes in a vertical hyperbola
m = (plus or minus) b/a
The formula for the slopes of the asymptotes in a horizontal hyperbola
Vertical (y is first)
Is this hyperbola vertical or horizontal? y^2/4 - x^2/16 = 1
Vertices are (5, 0) and (-5, 0)
What are the vertices in this hyperbola? x^2/25 - y^2/9 = 1
What are the slopes of the asymptotes? x^2/36 - y^2/100 = 1
This is a horizontal hyperbola so we use b/a which is 10/6 or 5/3 (+ and -)
Where are the foci in this hyperbola? y^2/36 - x^2/64 = 1
First, a^2 + b^2 = 100, so c = 10. Since this is a vertical hyperbola (y is first), the foci are (0, 10) and (0, -10)
Circle
r= asin(theta)
Parabola
the graph of a quadratic function
Ellipse
Equation adds x²/a² and y²/b²=1
Hyperbola
Equation subtracts x²/a and y²/b=1; x and y coefficients have different signs.
Line
A straight path that goes without end in two directions.
horizontal ellipse equation
(x-h)2/a2 + (y-k)2/b2 = 1
h+-a, k
horizontal ellipse & hyperbola vertices
h+-c, k
horizontal ellipse & hyperbola foci
c2=a2-b2
ellipse a2 b2 c2 equation
vertical ellipse equation
(x-h)2/b2 + (y-k)2/a2 = 1
h, k+-a
vertical ellipse & hyperbola vertices
h, k+-c
vertical ellipse & hyperbola foci
horizontal hyperbola equation
(x-h)2/a2 - (y-k)2/b2 = 1
y-k=+-b/a(x-h)
horizontal hyperbola asymptotes
c2=a2+b2
hyperbola a2 b2 c2 equation
vertical hyperbola equation
(y-k)2/a2 - (x-h)2/b2= 1
y-k=+-a/b(x-h)
vertical hyperbola asymptotes
Positive
Counting from the decimal point to the right makes the exponent ___________________________________________
bigger
ellipse a2 always...
circle equation
(x-h)2 + (y-k)2 = r2
(x-h)2 = 4p(y-k)
vertical parabola equation
h, k+p
vertical parabola focus
y=k-p
vertical parabola directrix
(y-k)2 = 4p(x-h)
horizontal parabola equation
h+p, k
horizontal parabola focus
x=h-p
horizontal parabola directrix
Ax²+By²+Cx+Dy+E=0
General equation for ellipse