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

Section 7.3 – The Ellipse
Ellipse – a set of points in a plane whose distances from two fixed points
is a constant.
Section 7.3 – The Ellipse
Ellipse – a set of points in a plane whose sum of the distances from two
fixed points is a constant.
𝑑 𝐹1 , 𝑃 + 𝑑 𝐹2 , 𝑃 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Q
𝑑 𝐹1 , 𝑄 + 𝑑 𝐹2 , 𝑄 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝑑 𝐹1 , 𝑃 + 𝑑 𝐹2 , 𝑃
Section 7.3 – The Ellipse
Foci – the two fixed points, 𝐹1 𝑎𝑛𝑑 𝐹2 , 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, 𝑉1 𝑎𝑛𝑑 𝑉2 .
Foci
Minor axis – the line that
is perpendicular to the
major axis and goes
through the center of
the ellipse.
Major axis
Minor axis
Vertices
Section 7.3 – The Ellipse
Section 7.3 – The Ellipse
Section 7.3 – The Ellipse
Find the vertices for the major and minor axes, and the foci using the
following equation of an ellipse.
𝑥2 𝑦2
+
=1
25 9
Major axis is along the x-axis
Vertices of major axis:
𝑎2 = 25
𝑎 = ±5
−5,0 𝑎𝑛𝑑 (5,0)

Vertices of the minor axis
𝑏 2 = 9 𝑏 = ±3
0,3 𝑎𝑛𝑑 (0, −3)
Foci
𝑐2
=
𝑎2
− 𝑏2
𝑐 2 = 16
𝑐2
= 25 − 9
𝑐 = ±4
−4,0 𝑎𝑛𝑑 (4,0)
 
 

Section 7.3 – The Ellipse
Find the vertices for the major and minor axes, and the foci using the
following equation of an ellipse.
4𝑥 2 + 9𝑦 2 = 36
4𝑥 2 9𝑦 2
+
=1
36
36
Major axis is along the x-axis
Vertices of major axis:
𝑎2 = 9
𝑎 = ±3
−3,0 𝑎𝑛𝑑 (3,0)
Vertices of the minor axis
𝑏 2 = 4 𝑏 = ±2
0,2 𝑎𝑛𝑑 (0, −2)
Foci
𝑐 2 = 𝑎2 − 𝑏 2
𝑐2 = 5
𝑥2 𝑦2
+
=1
9
4
𝑐2 = 9 − 4
𝑐=± 5
− 5, 0 𝑎𝑛𝑑 ( 5, 0)




Section 7.3 – The Ellipse
Find the equation of an ellipse given a vertex of 0,12 and a focus of
(−2 11). Graph the ellipse.
Vertices of major axis:
0,12 𝑎𝑛𝑑 (0, −12)
Vertices of the minor axis
𝑎 = ±12
𝑎2 = 144
𝑐 = ±2 11

𝑐 2 = 44

𝑏 2 = 𝑎2 − 𝑐 2
𝑏 2 = 100 𝑏 = ±10
−10,0 𝑎𝑛𝑑 (10,0)
𝑥2 𝑦2
+ 2=1
2
𝑏
𝑎


𝑏 2 = 144 − 44


𝑥2
𝑦2
+
=1
100 144
Section 7.3 – The Ellipse
Section 7.3 – The Ellipse
Find the center, vertices, and foci given the following equation of an
ellipse.
(𝑥 − 3)2 (𝑦 − 9)2
+
= 1 Center: (3,9)
25
9
Major axis is along the x-axis
Foci
Vertices: 𝑎2 = 25 𝑎 = ±5
𝑐 2 = 𝑎2 − 𝑏 2
3 − 5,9 𝑎𝑛𝑑 (3 + 5,9)
𝑐 2 = 25 − 9
−2,9 𝑎𝑛𝑑 (8,9)
𝑐 2 = 16
Vertices of the minor axis
𝑏 2 = 9 𝑏 = ±3
3,9 − 3 𝑎𝑛𝑑 (3,9 + 3)
3,6 𝑎𝑛𝑑 (3,12)
𝑐 = ±4
3 − 4,9 𝑎𝑛𝑑 (3 + 4,9)
−1,9 𝑎𝑛𝑑 (7,9)
Section 7.3 – The Ellipse
Find the center, vertices, and foci given the following equation of an
ellipse.
(𝑥 − 3)2 (𝑦 − 9)2
+
=1
25
9
Center:
(3,9)

Vertices:
−2,9 𝑎𝑛𝑑 (8,9)
Vertices of the minor axis
3,6 𝑎𝑛𝑑 (3,12)
Foci
−1,9 𝑎𝑛𝑑 (7,9)
 


 
Section 7.3 – The Ellipse
Find the center, the vertices of the major and minor axes, and the foci
using the following equation of an ellipse.
16𝑥 2 + 4𝑦 2 + 96𝑥 − 8𝑦 + 84 = 0
16𝑥 2 + 96𝑥 + 4𝑦 2 − 8𝑦 = −84
16(𝑥 2 + 6𝑥) + 4(𝑦 2 − 2𝑦) = −84
6
−2
2
=3 3 =9
= −1 (−1)2 = 1
2
2
16(𝑥 2 + 6𝑥 + 9) + 4 𝑦 2 − 2𝑦 + 1 = −84 + 144 + 4
16(𝑥 + 3)2 +4(𝑦 − 1)2 = 64
16(𝑥 + 3)2 4(𝑦 − 1)2
+
=1
64
64
(𝑥 + 3)2 (𝑦 − 1)2
+
=1
4
16
Section 7.3 – The Ellipse
(𝑥 + 3)2 (𝑦 − 1)2
+
=1
4
16
Center:
(−3,1)
Major axis: 𝑥 = −3 (𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙)
Vertices: 𝑎2 = 16 𝑎 = ±4
−3,1 − 4 𝑎𝑛𝑑 (−3,1 + 4)
−3, −3 𝑎𝑛𝑑 (−3,5)
Minor axis: 𝑦 = 1 (ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙)
Vertices of the minor axis
𝑏2 = 4
𝑏 = ±2
−3 − 2,1 𝑎𝑛𝑑 (−3 + 2,1)
−5,1 𝑎𝑛𝑑 (−1,1)
Foci
𝑐 2 = 𝑎2 − 𝑏 2
𝑐 2 = 16 − 4
𝑐 2 = 12
𝑐 = ±2 3
−3,1 − 2 3 𝑎𝑛𝑑 (−3,1 + 2 3)
−3, −2.464 𝑎𝑛𝑑 (−3, 4.464)
Section 7.3 – The Ellipse
(𝑥 + 3)2 (𝑦 − 1)2
+
=1
4
16
Center:
(−3,1)
Major axis vertices:
−3, −3 𝑎𝑛𝑑 (−3,5)
Minor axis vertices:
−5,1 𝑎𝑛𝑑 (−1,1)
Foci
−3, −2.464 𝑎𝑛𝑑 (−3,4.464)

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