conic sections definition: a conic section is the intersection of a plane and a cone
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Conic Sections
Definition: A conic section is the intersection of a plane and a cone
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Identifying Conic Sections
How do I determine whether the graph of an equation represents a conic, and if so, which conic does it represent, a circle, an ellipse, a parabola or a hyperbola?
Created by K. Chiodo, HCPS
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General Form of a Conic Equation
We usually see conic equations written in General, or Implicit Form:
Ax2 +Bxy+Cy2 +Dx+Ey+F =0where A, B, C, D, E and F are integers and A, B and C are NOT ALL equal to zero.
Note: You may see some conic equations solved for y, but if the equation can be re-written into the form above, it is a conic equation!
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Please Note:A conic equation written in General Form doesn’t have to have all SIX terms! Several of the coefficients A, B, C, D, E and F can equal zero, as long as A, B and C don’t ALL equal zero.
Dx+Ey+F =0
Linear!
If A, B and C all equal zero, what kind of equation do you have?
T H I N K......
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So, it’s a conic equation if...
• the highest degree (power) of x and/or y is 2 (at least ONE has to be squared)
• the other terms are either linear, constant, or the product of x and y
• there are no variable terms with rational exponents (i.e. no radical expressions) or terms with negative exponents (i.e. no rational expressions)
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What values form an Ellipse?
The values of the coefficients in the conic equation determine the TYPE of conic.
Ax2+Bxy+Cy2+Dx+Ey+F =0
What values form a Hyperbola?
What values form a Parabola?
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Ellipses...
Ax2 +Cy2 +Dx+Ey+F =0
NOTE: There is no Bxy term, and D, E & F may equal zero!
where A & C have the SAME SIGN
For example: 2x2 +y2 +8x=0
2x2 +2y2 +8x−6=0
−x2 −2x−3y2 +6y=0
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Ellipses...The General, or Implicit, Form of the equations can be converted to Graphing Form by completing the square and dividing so that the constant = 1.
2x2 +y2 +8x=0
2(x2 +4x+4)+y2 =8
2(x+2)2 +y2 =8
2(x+2)2
8+y2
8=
88
(x+2)2
4+y2
8=1
This is an ellipse since x & y are both squared, and both quadratic terms have the same sign!
Center (-2, 0)
Vert. Axis = √8Hor. Axis = 2
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Ellipses...In this example, x2 and y2 are both negative (still the same sign), you can see in the final step that when we divide by negative 4 all of the terms are positive.
−x2 −2x−3y2 +6y=0
−(x2 +2x+1) −3(y2 −2y+1) =−1−3
−(x+1)2 −3(y−1)2 =−4
−(x+1)2
−4−
3(y−1)2
−4=1
(x+1)2
4+
(y−1)2
43
=1
Vert. axis = 2/√3
Hor. axis = 2
center (-1, 1)
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Ellipses…a special case!
it is a ...
When A & C are the same value as well as the same
sign, the ellipse is the same length in all directions …
2x2 +2y2 +8x−6=0
2(x2 +4x+2) +2y2 =6+4
2(x+2)2 +2y2 =10
(x+2)2
5+y2
5=1
Circle!
Center (-2, 0)
Radius = √5
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Hyperbola...
Ax2 +Cy2 +Dx+Ey+F =0
NOTE: There is no Bxy term, and D, E & F may equal zero!
where A & C have DIFFERENT signs.
For example: 9x2 −4y2 −36x−8y−4=0x2 −y2 +6y−5=0x2 +10x−4y2 +8y+5=0
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Hyperbola...
The General, or Implicit, Form of the equations can be converted to Graphing Form by completing the square and dividing so that the constant = 1.
9x2 −4y2 −36x−8y−4=0
9(x2 −4x)−4(y2 +2y) =4
9(x2 −4x+4) −4(y2 +2y+1) =4+36−4
9(x−2)2 −4(y+1)2 =36
9(x−2)2
36−
4(y+1)2
36=
3636
(x−2)2
4−
(y+1)2
9=1
This is a hyperbola since x & y are both squared, and the quadratic terms have different signs!
Center (2,-1)
y-axis=3x-axis=2
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Hyperbola...
In this example, the signs change, but since the quadratic terms still have different signs, it is still a hyperbola!
x2 −y2 +6y−5=0
x2 −(y2 −6y) =5
x2 −(y2 −6y+9) =5−9
x2 −(y−3)2 =−4
x2
−4−
(y−3)2
−4=
−4−4
(y−3)2
4−x2
4=1
Center (0,3)x-axis=2
y-axis=2
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Parabola... A Parabola can be
oriented 2 different ways:
Ax2 +Dx+Ey+F =0
A parabola is vertical if the equation has an x squared term AND a linear y term; it may or may not have a linear x term & constant:
Cy2 +Dx+Ey+F =0
A parabola is horizontal if the equation has a y squared term AND a linear x term; it may or may not have a linear y term & constant:
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Parabola …Vertical
x2 −4x−y+7=0
The following equations all represent vertical parabolas in general form; they all have a squared x term and a linear y term:
4x2 +8x+y=0
x2 −y−7=0
x2 +y=0
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Parabola …VerticalTo write the equations in Graphing Form, complete the square for the x-terms. There are 2 popular conventions for writing parabolas in Graphing Form, both are given below:
Vertex (2,3)
0=x2 −4x−y+7
0=(x2 −4x+4)−y+7−4
0=(x−2)2 −y+3
y=(x−2)2 +3
or
y−3=(x−2)2
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Parabola …VerticalIn this example, the signs must be changed at the end so that the y-term is positive, notice that the negative coefficient of the x squared term makes the parabola open downward.
Vertex (-1,4)0=4x2 +8x+y
0=4(x2 +2x+1)+y−4
0=4(x+1)2 +y−4
−y=4(x+1)2 −4
y=−4(x+1)2 +4
or
y−4=−4(x+1)2
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Parabola …Horizontal
y2 +8y−2x+18=0
The following equations all represent horizontal parabolas in general form, they all have a squared y term and a linear x term:
x+y2 −3=03y2 −6y+x−2=0
y2 −x=0
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Parabola …HorizontalTo write the equations in Graphing Form, complete the square for the y-terms. There are 2 popular conventions for writing parabolas in Graphing Form, both are given below:
0=y2 +8y−2x+18
0=(y2 +8y+16) −2x+18−16
0=(y+4)2 −2x+2
2x=(y+4)2 +2
x=12
(y+4)2 +1
or
0=(y+4)2 −2(x−1)
2(x−1) =(y+4)2
Vertex (1,-4)
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Parabola …Horizontal
0 =x+y2 −3
0 =y2 +x−3
−x=y2 −3
x=−y2 +3
or
(x−3) =−y2
In this example, the signs must be changed at the end so that the x-term is positive; notice that the negative coefficient of the y squared term makes the parabola open to the left.
Vertex (3,0)
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What About the term Bxy?
Ax2 +Bxy+Cy2 +Dx+Ey+F =0
None of the conic equations we have looked at so far included the term Bxy. This term leads to a hyperbolic graph:
4xy−8=0
y=84x
=2x
or, solved for y:
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Summary ...General Form of a Conic Equation:
Ax2 +Bxy+Cy2 +Dx+Ey+F =0where A, B, C, D, E and F are integers and A, B and C are NOT ALL equal to zero.
Identifying a Conic Equation: Conic Equation Stats
A = 0 or C = 0, but not both. Parabola If A = 0, then the parabola is horizontal.
If C = 0, then the parabola is vertical.
Circle A = C Ellipse A & C have the same sign. Hyperbola A & C have different signs.
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Practice ...Identify each of the following equations as a(n):
(a) ellipse (b) circle (c) hyperbola
(d) parabola (e) not a conic
See if you can rewrite each equation into its Graphing Form!1) x2 +4y2 +2x−24y+33=0
2) 4x2 −4y2 −9=0
3) x2 −4x−y=0
4) x2 +y2 −2x−8=0
5) 9x2 +25y2 −54x−50y−119=0
6) x2 −x=0
7) y2 −8y−9x+52=0
8) x2 −2x−y2 +4y−7=0
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Answers ...1) x2 +4y2 +2x−24y+33=0 - -- > (a)
(x+1)2
4+
(y−3)2
1=1
2) 4x2 −4y2 −9=0 - -- > (c) x2
94
−y2
94
=1
3) x2 −4x−y=0 - --> (d) (y+4) =(x−2)2
4) x2 +y2 −2x−8=0 - -- > (b) (x−1)2 +y2 =9
5) 9x2 +25y2 −54x−50y−119=0 - - > (a) (x−3)2
25+
(y−1)2
9=1
6) x2 −x=0 - --> (e) not a conic
7) y2 −8y−9x+52=0 --- > (d) 9(x−4) =(y−4)2
8) x2 −2x−y2 +4y−7=0 - -- > (c) (x−1)2
4−
(y−2)2
4=1
(a) ellipse (b) circle (c) hyperbola (d) parabola (e) not a conic
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Conic Sections !
CE
Created by K. Chiodo, HCPS
H
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