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Hyperbola – a set of points in a plane whose difference of the distances from two fixed points is a constant. Section 7.4 – The Hyperbola
Hyperbola – a set of points in a plane whose difference of the distances from two fixed points is a constant.
Section 7.4 – The Hyperbola
Section 7.4 – The Hyperbola
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
Section 7.4 – The HyperbolaFoci – 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
Section 7.4 – The Hyperbola
Section 7.4 – The Hyperbola
Section 7.4 – The Hyperbola
Section 7.4 – The Hyperbola
(−𝑏 ,0) (𝑏 ,0)
Section 7.4 – The Hyperbola
Section 7.4 – The HyperbolaIdentify the direction of opening, the coordinates of the center, the vertices, and the foci. Find the equations of the asymptotes and sketch the graph.
𝑥2
16− 𝑦2
9=1
Vertices of transverse axis:𝑎2=16
Center:
Equations of the AsymptotesFoci𝑏2=9𝑏2=𝑐2−𝑎2
𝑎=±4(−4,0 )𝑎𝑛𝑑 (4,0)
𝑏=±3 (0,3 )𝑎𝑛𝑑(0 ,−3)9=𝑐2−16
𝑐2=25 𝑐=±5(−5,0 )𝑎𝑛𝑑(5,0)
𝑦− 𝑦1=±𝑏𝑎 (𝑥−𝑥1)
𝑦−0=± 34 (𝑥−0)
𝑦=± 34 𝑥
Section 7.4 – The Hyperbola
𝑦2
4− 𝑥2
16=1
Vertices of transverse axis:𝑎2=4
Center:
Equations of the AsymptotesFoci𝑏2=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=± 12 𝑥
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.
Section 7.4 – The Hyperbola
(𝑦−𝑘)2
𝑎2 −(𝑥−h)2
𝑏2 =1
Find b:𝑎2=1
Center:
𝑏2=9−1=8𝑏2=𝑐2−𝑎2
𝑎=4−3=1
Equation of the Hyperbola
𝑏=±2√2(−4−2√2 ,0 )𝑎𝑛𝑑 (−4+2√2 ,0)
𝑐2=9𝑐=3−0=3
(−6.83,3 )𝑎𝑛𝑑(−1.17,3)
A hyperbola has a focus at and vertices at . What is its equation? Graph the hyperbola.
(−4+(−4)2
, 4+22 )(−4,3)
(𝑦−3)2
1−
(𝑥+4)2
8=1
Section 7.4 – The Hyperbola
Center:
Equations of the Asymptotes
𝑎=1𝑏=±2√2
𝑦− 𝑦1=±𝑎𝑏 (𝑥−𝑥1)
𝑦−3=± 12√2
(𝑥+4)
A hyperbola has a focus at and vertices at . What is its equation? Graph the hyperbola.
(−4,3)
𝑦−3=± √24
(𝑥+4)
Section 7.4 – The HyperbolaFind 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=25
182 =992=81
( 𝑦2+10 𝑦+25 )−4 (𝑥¿¿2+18 𝑥+81)=399+25−3 24¿(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
Section 7.4 – The HyperbolaFind 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(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
Section 7.4 – The HyperbolaFind 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
(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)
