Ellipses ppt

1. Section 10.2 – The Ellipse
Ellipse – a set of points in a plane whose distances from two fixed
points is a constant.

2. Section 10.2 – The Ellipse
Ellipse – a set of points in a plane whose sum of the distances from two
fixed points is a constant.

Q
𝑑 𝐹1, 𝑃 + 𝑑 𝐹2, 𝑃 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑑 𝐹1, 𝑄 + 𝑑 𝐹2, 𝑄 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝑑 𝐹1, 𝑃 + 𝑑 𝐹2, 𝑃

3. Section 10.2 – 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 .
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

4. Section 10.2 – The Ellipse

5. Section 10.2 – The Ellipse

6. Section 10.2 – 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
Foci
𝑏2 = 9
𝑐2 = 𝑎2 − 𝑏2
𝑎 = ±5 −5,0 𝑎𝑛𝑑 (5,0)
𝑏 = ±3 0,3 𝑎𝑛𝑑 (0, −3)
𝑐2 = 25 − 9
𝑐2 = 16 𝑐 = ±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.
 


 

7. Section 10.2 – The Ellipse
4𝑥2
36
+
9𝑦2
36
= 1
Vertices of major axis:
𝑎2 = 9
Major axis is along the x-axis
Vertices of the minor axis
Foci
𝑏2 = 4
𝑐2 = 𝑎2 − 𝑏2
𝑎 = ±3 −3,0 𝑎𝑛𝑑 (3,0)
𝑏 = ±2 0,2 𝑎𝑛𝑑 (0, −2)
𝑐2 = 9 − 4
𝑐2
= 5 𝑐 = ± 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.
 


 
4𝑥2
+ 9𝑦2
= 36
𝑥2
9
+
𝑦2
4
= 1

8. Section 10.2 – The Ellipse
𝑥2
𝑏2
+
𝑦2
𝑎2
= 1
Vertices of major axis:
𝑎2 = 144
Vertices of the minor axis
𝑏2
= 100
𝑏2 = 𝑎2 − 𝑐2
𝑎 = ±12
−10,0 𝑎𝑛𝑑 (10,0)
𝑏 = ±10
0,12 𝑎𝑛𝑑 (0, −12)
𝑐2
= 44
𝑐 = ±2 11
Find the equation of an ellipse given a vertex of 0,12 and a focus of
(0, −2 11). Graph the ellipse.
 




𝑏2
= 144 − 44
𝑥2
100
+
𝑦2
144
= 1

9. Section 10.2 – The Ellipse

10. Section 10.2 – The Ellipse
(𝑥 − 3)2
25
+
(𝑦 − 9)2
9
= 1
Vertices: 𝑎2 = 25
Major axis is along the x-axis
Vertices of the minor axis
Foci
𝑏2 = 9
𝑐2
= 𝑎2
− 𝑏2
𝑎 = ±5
3 − 5,9 𝑎𝑛𝑑 (3 + 5,9)
𝑏 = ±3
3,9 − 3 𝑎𝑛𝑑 (3,9 + 3)
𝑐2 = 25 − 9
𝑐2 = 16
𝑐 = ±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)

11. Section 10.2 – 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)


12. Section 10.2 – The Ellipse
Find the center, the vertices of the major and minor axes, and the foci
using the following equation of an ellipse.
16𝑥2
+ 4𝑦2
+ 96𝑥 − 8𝑦 + 84 = 0
16𝑥2 + 96𝑥 + 4𝑦2 − 8𝑦 = −84
16(𝑥2 + 6𝑥) + 4(𝑦2 − 2𝑦) = −84
6
2
= 3 32 = 9
−2
2
= −1 (−1)2= 1
16(𝑥2 + 6𝑥 + 9) + 4 𝑦2 − 2𝑦 + 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

13. Section 10.2 – The Ellipse
Center:
(−3,1)
(𝑥 + 3)2
4
+
(𝑦 − 1)2
16
= 1
Major axis: 𝑥 = −3 (𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙)
Vertices: 𝑎2 = 16
Vertices of the minor axis
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 − 4
𝑐2 = 12
𝑐 = ±2 3
−3,1 − 2 3 𝑎𝑛𝑑 (−3,1 + 2 3)
−3, −2.464 𝑎𝑛𝑑 (−3, 4.464)
Minor axis: 𝑦 = 1 (ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙)

14. Section 10.2 – 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)