Section 7.3 – The EllipseEllipse – a set of points in a plane whose distances from two fixed points is a constant.

Section 7.3 – The EllipseEllipse – 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 )

Section 7.3 – The EllipseFoci – 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

Section 7.3 – The Ellipse

Section 7.3 – The Ellipse

Section 7.3 – The Ellipse

𝑥2

25+𝑦2

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.

Section 7.3 – The Ellipse

4 𝑥2

36+

9 𝑦2

36=1

Vertices of major axis:𝑎2=9

Major axis is along the x-axis

Vertices of the minor 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

Section 7.3 – The Ellipse

𝑥2

𝑏2 +𝑦2

𝑎2 =1

Vertices of major axis:

𝑎2=144Vertices of the minor axis

𝑏2=100

𝑏2=𝑎2−𝑐2

𝑎=±12

(−10,0 )𝑎𝑛𝑑(10,0)𝑏=±10

(0,12 )𝑎𝑛𝑑 (0 ,−12)

𝑐2=44𝑐=±2√11

Find the equation of an ellipse given a vertex of and a focus of . Graph the ellipse.

𝑏2=144−44

𝑥2

100+𝑦2

144=1

Section 7.3 – The Ellipse

Section 7.3 – The Ellipse

(𝑥−3)2

25+(𝑦−9)2

9=1

Vertices:𝑎2=25Major axis is along the x-axis

Vertices of the minor axis

Foci

𝑏2=9

𝑐2=𝑎2−𝑏2𝑎=±5(3−5,9 )𝑎𝑛𝑑 (3+5,9)

𝑏=±3(3,9−3 )𝑎𝑛𝑑(3,9+3)

𝑐2=25−9𝑐2=16𝑐=±4

(3−4,9 )𝑎𝑛𝑑 (3+4,9)

Find the center, vertices, and foci given the following equation of an ellipse.

Center:(3,9)

(−2,9 )𝑎𝑛𝑑(8,9)

(3,6 )𝑎𝑛𝑑(3,12)(−1,9 )𝑎𝑛𝑑 (7,9)

Section 7.3 – The Ellipse

(𝑥−3)2

25+(𝑦−9)2

9=1

Find the center, vertices, and foci given the following equation of an ellipse.

Center:(3,9)Vertices:

Vertices of the minor axis

Foci

(−2,9 )𝑎𝑛𝑑(8,9)

(3,6 )𝑎𝑛𝑑(3,12)

(−1,9 )𝑎𝑛𝑑 (7,9)

Section 7.3 – The EllipseFind 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

Section 7.3 – The Ellipse

Center:(−3,1)

(𝑥+3)2

4+

(𝑦−1)2

16=1

Major axis: Vertices:𝑎2=16

Vertices of the minor axis𝑏2=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)

Foci𝑐2=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:

Section 7.3 – The Ellipse

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 )