Conics – curves that are created by the intersection of a plane and a right circular cone. Section...

Conics – curves that are created by the intersection of a plane and a right circular cone.

Section 11.6 – Conic Sections

Parabola – set of points in a plane that are equidistant from a fixed point (d(F, P)) and a fixed line (d (P, Q)).

Focus - the fixed point of a parabola.

Directrix - the fixed line of a parabola. Axis of Symmetry

Directrix

Axis of Symmetry – The line that goes through the focus and is perpendicular to the directrix.

Vertex – the point of intersection of the axis of symmetry and the parabola.


Section 11.6 – Conic SectionsParabolas

𝑦 2=4𝑝𝑥 𝑥2=4𝑝𝑦

(𝑥−h)2=4𝑝 (𝑦−𝑘)(𝑦−𝑘)2=4𝑝 (𝑥−h)

Find the vertex, focus and the directrix

𝑥2=16 𝑦

𝑣 𝑒𝑟𝑡𝑒𝑥 :(0,0)

16=4𝑝𝑝=4

𝑓 𝑖𝑛𝑑𝑝

𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎 ,𝑜𝑝𝑒𝑛𝑠𝑢𝑝

𝑓 𝑜𝑐𝑢𝑠(0,0+𝑝)(0,4 )

𝑦=0−𝑝

𝑑𝑖𝑟𝑒𝑐𝑡𝑟𝑖𝑥

𝑦=−4

𝑦=−4


Find the vertex and the focus given:

1=4𝑝

𝑝=14

𝑓 𝑖𝑛𝑑𝑝

𝑜𝑝𝑒𝑛𝑠𝑙𝑒𝑓𝑡

𝑣 𝑒𝑟𝑡𝑒𝑥

𝑓 𝑜𝑐𝑢𝑠(5−

14,−5)

𝑦 2+10 𝑦+𝑥+20=0

𝑐 𝑜𝑚𝑝𝑙𝑒𝑡𝑒 h𝑡 𝑒𝑠𝑞𝑢𝑎𝑟𝑒𝑦 2+10 𝑦=−𝑥−20

102

=5 52=25

𝑦 2+10 𝑦+25=−𝑥−20+25(𝑦+5)2=−𝑥+5(𝑦+5)2=−(𝑥−5)

(5 ,−5)

(434,−5)

𝑑𝑖𝑟𝑒𝑐𝑡𝑟𝑖𝑥𝑥=5+

14

𝑥=514

𝑥=514


Ellipse – a set of points in a plane whose sum of the distances from two fixed points is a constant.

Q

𝑑 (𝐹 1 ,𝑃 )+𝑑 (𝐹2 ,𝑃 )=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑑 (𝐹 1 ,𝑄 )+𝑑 (𝐹 2 ,𝑄 )=¿𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡¿𝑑 (𝐹 1 ,𝑃 )+𝑑 (𝐹 2 ,𝑃 )


Foci – the two fixed points, , whose distances from a single point on the ellipse is a constant.

Major axis – the line that contains the foci and goes through the center of the ellipse.

Vertices – the two points of intersection of the ellipse and the major axis, .

Minor axis – the line that is perpendicular to the major axis and goes through the center of the ellipse.

Foci

Major axis

Vertices

Minor axis


Equation of an Ellipse Centered at the Origin


𝑥2

𝑎2 +𝑦2

𝑏2 =1 h𝑤 𝑒𝑟𝑒𝑎>𝑏 𝑥2

𝑏2 +𝑦2

𝑎2 =1 h𝑤 𝑒𝑟𝑒𝑎>𝑏

Section 11.6 – Conic SectionsEquation of an Ellipse Centered at a Point

𝑥2

25+ 𝑦

2

9=1

Vertices of major axis:

𝑎2=25

Major axis is along the x-axis

Vertices of the minor axis

Foci𝑏2=9

𝑐2=𝑎2−𝑏2

𝑎=±5 (−5,0 )𝑎𝑛𝑑(5,0)

𝑏=±3 (0,3 )𝑎𝑛𝑑(0 ,−3)

𝑐2=25−9𝑐2=16𝑐=±4

(−4,0 )𝑎𝑛𝑑 (4,0)

Find the vertices for the major and minor axes, and the foci using the following equation of an ellipse.


4 𝑥2

36+ 9 𝑦2

36=1

Vertices of major axis:

𝑎2=9

Major axis is along the x-axis


Foci𝑏2=4

𝑐2=𝑎2−𝑏2

𝑎=±3(−3,0 )𝑎𝑛𝑑(3,0)

𝑏=±2 (0,2 )𝑎𝑛𝑑 (0 ,−2)

𝑐2=9−4𝑐2=5𝑐=±√5

(−√5 ,0 )𝑎𝑛𝑑 (√5 ,0)

Find the vertices for the major and minor axes, and the foci using the following equation of an ellipse.

4 𝑥2+9 𝑦2=36𝑥2

9+ 𝑦2

4=1


Find the center, the vertices of the major and minor axes, and the foci using the following equation of an ellipse.

16 𝑥2+4 𝑦2+96 𝑥−8 𝑦+84=016 𝑥2+96 𝑥+4 𝑦2−8 𝑦=−8416 (𝑥¿¿ 2+6 𝑥)+4 (𝑦2−2 𝑦 )=−84 ¿

62=332=9

−22

=−1(−1)2=1

16 (𝑥¿¿ 2+6 𝑥+9)+4 (𝑦 2−2 𝑦+1 )=−84+144+4¿16 (𝑥+3)2+4 (𝑦−1)2=64

16(𝑥+3)2

64+

4 (𝑦−1)2

64=1

(𝑥+3)2

4+

(𝑦−1)2

16=1


Center:

(−3,1)

(𝑥+3)2

4+

(𝑦−1)2

16=1

Major axis: Vertices:𝑎2=16


𝑏2=4

𝑎=±4(−3,1−4 )𝑎𝑛𝑑(−3,1+4 )

𝑏=±2(−3−2,1 )𝑎𝑛𝑑(−3+2,1)

(−3 ,−3 )𝑎𝑛𝑑 (−3,5)

(−5,1 )𝑎𝑛𝑑 (−1,1)

Foci

𝑐2=𝑎2−𝑏2

𝑐2=16−4𝑐2=12𝑐=±2√3

(−3,1−2√3 )𝑎𝑛𝑑 (−3,1+2√3)(−3 ,−2.464 )𝑎𝑛𝑑 (−3 ,4.464)

Minor axis:


Center:

(−3,1)

(𝑥+3)2

4+

(𝑦−1)2

16=1

Major axis vertices:

Minor axis vertices:(−5,1 )𝑎𝑛𝑑 (−1,1)

(−3 ,−3 )𝑎𝑛𝑑 (−3,5)

Foci

(−3 ,−2.464 )𝑎𝑛𝑑 (−3,4.464 )


Q

𝑑 (𝐹 1 ,𝑄 )−𝑑 (𝐹 2,𝑄 )=¿𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡¿±2𝑎

Hyperbola – a set of points in a plane whose difference of the distances from two fixed points is a constant.

𝑑 (𝐹 1 ,𝑃 )−𝑑 (𝐹2 ,𝑃 )=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡=±2𝑎


Foci – the two fixed points, , whose difference of the distances from a single point on the hyperbola is a constant.

Transverse axis – the line that contains the foci and goes through the center of the hyperbola.

Vertices – the two points of intersection of the hyperbola and the transverse axis, .

Conjugate axis – the line that is perpendicular to the transverse axis and goes through the center of the hyperbola.

Conjugate axis

Center – the midpoint of the line segment between the two foci.

Center


Equation of an Ellipse Centered at the Origin


Equation of a Hyperbola Centered at the Origin


Equation of a Hyperbola Centered at a Point


𝑦2

4−𝑥2

16=1

Vertices of transverse axis:

𝑎2=4

Center:

Equations of the AsymptotesFoci

𝑏2=16𝑏2=𝑐2−𝑎2

𝑎=±2(0 ,−2 )𝑎𝑛𝑑(0,2)

𝑏=±4 (−4,0 )𝑎𝑛𝑑 (4,0)16=𝑐2−4

𝑐2=20 𝑐=±2√5( 0 ,−2√5 )𝑎𝑛𝑑(0,2√5)

𝑦− 𝑦1=±𝑎𝑏

(𝑥−𝑥1)

𝑦−0=±24(𝑥−0)

𝑦=±12𝑥

Identify the direction of opening, the coordinates of the center, the vertices, and the foci. Find the equations of the asymptotes and sketch the graph.


Find the center, the vertices of the transverse axis, the foci and the equations of the asymptotes using the following equation of a hyperbola.

𝑦 2−4 𝑥2−72𝑥+10 𝑦−399=0

( 𝑦2+10 𝑦 )−4 (𝑥¿¿2+18 𝑥)=399¿102

=552=25182

=992=81

( 𝑦2+10 𝑦+25 )−4 (𝑥¿¿2+18 𝑥+81)=399+25−324¿(𝑦+5)2−4 (𝑥+9)2=100

(𝑦+5)2

100−

4(𝑥+9)2

100=1

(𝑦+5)2

100−

(𝑥+9)2

25=1

𝑦 2+10 𝑦−4 𝑥2−72 𝑥=399

Opening up/down



𝑦 2−4 𝑥2−72𝑥+10 𝑦−399=0(𝑦+5)2

100−

(𝑥+9)2

25=1

Center:

Vertices:

𝑎2=100 𝑎=10(−9 ,−5−10 )𝑎𝑛𝑑 (−9 ,−5+10)

(−9 ,−15 )𝑎𝑛𝑑(−9,5)

Foci:

25=𝑐2−100𝑏2=𝑐2−𝑎2

(−9 ,−5−5√5 )𝑎𝑛𝑑(−9 ,−5+5√5)𝑐=√125=5√5

(−9 ,−16.18 )𝑎𝑛𝑑(−9,6.18)

𝑐2=125



𝑦 2−4 𝑥2−72𝑥+10 𝑦−399=0(𝑦+5)2

100−

(𝑥+9)2

25=1

Center:

𝑎=10

Equations of the Asymptotes

𝑦− 𝑦1=±𝑎𝑏

(𝑥−𝑥1)

𝑏=5

𝑦−(−5)=±105

(𝑥−(−9))

𝑦+5=±2(𝑥+9)


Conics – curves that are created by the intersection of a plane and a right circular cone. Section...

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Transcript of Conics – curves that are created by the intersection of a plane and a right circular cone. Section...