1 DEFINITION OF A ELLIPSE ELLIPSES PROBLEM 4 PROBLEM 1 PROBLEM 3 PROBLEM 2 STANDARD FORMULAS FOR ELLIPSES PROBLEM 5 Standards 4, 9, 16, 17 END SHOW PRESENTATION

1

DEFINITION OF A ELLIPSE

ELLIPSES

PROBLEM 4

PROBLEM 1

PROBLEM 3

PROBLEM 2

STANDARD FORMULAS FOR ELLIPSES

PROBLEM 5

Standards 4, 9, 16, 17

END SHOW

PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

http://www.mrperezonlinemathtutor.com/

2

STANDARD 4:

Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes

STANDARD 9:

Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b) + c.

STANDARD 16:

Students demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it.

STANDARD 17:

Given a quadratic equation of the form ax + by + cx + dy + e = 0, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation.

2

2 2

ALGEBRA II STANDARDS THIS LESSON AIMS:



ESTÁNDAR 4:

Los estudiantes factorizan polinomios representando diferencia de cuadrados, trinomios cuadrados perfectos, y la suma de diferencia de cubos.

ESTÁNDAR 9:

Los estudiantes demuestran y explican los efectos que tiene el cambiar coeficientes en la gráfica de funciones cuadráticas; esto es, los estudiantes determinan como la gráfica de una parábola cambia con a, b, y c variando en la ecuación y=a(x-b) + c

ESTÁNDAR 16:

Los estudiantes demuestran y explican cómo la geometría de la gráfica de una sección cónica (ej. Las asimptótes, focos y excentricidad) dependen de los coeficientes de la ecuación cuadrática que las representa.

Estándar 17:

Dada una ecuación cuadrática de la forma ax +by + cx + dy + e=0, los estudiantes pueden usar el método de completar al cuadrado para poner la ecuación en forma estándar y pueden reconocer si la gráfica es un círculo, elipse, parábola o hipérbola. Los estudiantes pueden graficar la ecuación

2

22



4

ELLIPSE

Definition of Ellipse:

A ellipse is the set of all points in a plane such that the sum of the distances from the foci is constant.

d4d2d3 d5

d1

d6

y

x

d1 d2

d3 d4

d5 d6+

+

+

= k

= k

= kFocus 2Focus 1

Ellipse




STANDARD EQUATION OF AN ELLIPSE

•Ellipse with center at (h,k) with horizontal axis

has equation

In this case, major axis is horizontal.

•Ellipse with center at (h,k) with vertical axis

has equation

In this case, major axis is vertical.

b

F1(-c, 0)

ac

c

b

a

a = b + c2 2 2

a = b + c2 2 2

y

x

y

x

Minor axis

Major axis

Major axis

Minor axis

F2(c, 0)

(0,b)

(-a,0) (a,0)

(0,-b)

(0,a)

(b,0)

(0,-a)

(-b,0)

F2(0, c)

F1(0,-c)NOTE: These two ellipses are graphed with center (0,0)

(x – h) (y – k)2 2= 1

a2 b2+

(x – h) (y – k)2 2= 1

a2b2 +

a>b in both cases




6

42 6-2-4-6

2

4

6

-2

-4

-6

8 10-8-10

8

-8

10

x

yFind the foci, a, b, and c and the equation of the ellipse below:

(-4,-1)

(-4,2)

(1,-1)

We can see that a=5 and b=3

3

5

h= -4k= -1

a = b + c2 2 2

-b -b2 2

c = a - b2 22

c = 25-92

c = 162

a = 252 b = 92

c= 4

Focus 1 = ( h + c, k)

Focus 2 = ( h - c, k)

Focus 1 = ( -4+ , -1)

Focus 2 = (-4 - , -1)

Focus 1 = ( 0, -1)

Focus 2 = ( -8,-1)

4

4

The equation is:

(x-(-4)) (y-(-1))2 2

=125 9

+

(x – h) (y – k)2 2= 1

a2 b2+

(x+4) (y+1)2 2

=125 9

+




7

(x-3) (y-4)2 2

=125 9

+

(x-(+3)) (y-(+4))2 2

=125 9

+

(x – h) (y – k)2 2= 1

a2 b2+

h= 3

k= 4

a = 252

b = 92

a = b + c2 2 2

-b -b2 2

c = a - b2 22

c = 25-92

c = 162

a = 252

b = 92

a = 5

b = 3

Focus 1 = ( h + c, k)

Focus 2 = ( h - c, k)

Focus 1 = ( 3+ , 4)

Focus 2 = ( 3 - , 4)

Focus 1 = ( 7, 4)

Focus 2 = ( -1,4)

Graph the following ellipse equation:

(x-3) (y-4)2 2

=125 9

+

c= 4

4

4

Rewriting the equation to graph it:

25 > 9 So, this ellipse is horizontal.




8

Summarizing obtained information about ellipse to graph it:

a = 5b = 3

Focus 1 = ( 7, 4)

Focus 2 = ( -1, 4)

h= 3

k= 4

42 6-2-4-6

2

4

6

-2

-4

-6

8 10-8-10

8

-8

10

x

y

Major axis = 2aMinor axis = 2b

Center (3,4)




9

(x+2) (y-6)2 2

=164 81

+

(x-(-2)) (y-(+6))2 2

=164 81

+

h= -2

k= 6

a = 812

b = 642

a = b + c2 2 2

-b -b2 2

c = a - b2 22

c = 81-642

c = 172

a = 812

b =642

a = 9

b = 8

Focus 1 = ( h, k + c)

Focus 2 = ( h, k - c)

Focus 1 = ( -2, 6 + ) 17

17Focus 2 = (-2, 6 - )

Focus 1 = ( -2, 6 + 4.1)

Focus 1 = ( -2, 10.1)

Focus 2 = ( -2, 6 – 4.1)Focus 2 = ( -2, 1.9)

(x – h) (y – k)2 2= 1

b2 a2+

c = 17

c 4.1

81 > 64 So, this ellipse is vertical.

Graph the ellipse: (x+2) (y-6)2 2

=164 81

+




10


a = 9b = 8

Focus 1 = (-2, 10.1)

Focus 2 = (-2, 1.9)

h= -2

k= 6


Center (-2,6) y

42 6-2-4-6

2

4

6

8 10-8-10

8

10

x

12

14

16

18




11

16x + 64y -64x -384y + 384 = 0 2 2

16x -64x + 64y -384y + 384 = 02 2

16x -16(4)x + 64y -64(6)y + 384 = 02 2

16 x - 4x + + 64 y - 6y + + 384 = 16 + 642 242

2 42

262

2 62

2

16 x - 4x + + 64 y - y + + 384 = 16 + 642 2 (2)2

(2)2 (3)

2(3)

2

4 (4)9 (9)16 x - 4x + + 64 y - 6y + + 384 = 16 + 642 2

16(x-2) + 64(y-3) + 384 = 64 +5762 2

16(x-2) + 64(y-3) + 384 = 6402 2

-384 -384

16(x-2) + 64(y-3) = 2562 2

256 256

We know that 16x + 64y -64x -384y + 384 = 0 is an ellipse. Put it in the standard form find a, b, c and the foci and graph it.(x – h) (y – k)2 2

= 1a2 b2

2 2

+




12

16(x-2) + 64(y-3) = 2562 2

256 256

16(x-2) 64(y-3)2 2

=1256 256

+

16(x-2) 64(y-3)2 2

=1256 25616

16 64

64

+

(x-2) (y-3)2 2

=116 4

+

(x-(+2)) (y-(+3))2 2

=116 4

+

We know that 16x + 64y -64x -384y + 384 = 0 is an ellipse. Put it in the standard form find a, b, c and the foci and graph it.

2 2

(x – h) (y – k)2 2= 1

a2 b2+

(x – h) (y – k)2 2= 1

a2 b2+

h= 2

k= 3

a = 162

b = 42

a = b + c2 2 2

-b -b2 2

c = a - b2 22

c = 16-42

c = 122

12 26 23 31

2 3 = 122

c = 2 32

c = 2 32

c = 2 3

a = 162

b = 42

a = 4

b = 2

Focus 1 = ( h + c, k)

Focus 2 = ( h - c, k)

3.5

Focus 1 = ( 2+ , 3)2 3

2 3Focus 2 = ( 2 - , 3)

Focus 1 = ( 2 + 3.5, 3)

Focus 1 = ( 5.5, 3)

Focus 2 = ( 2 - 3.5, 3)Focus 2 = ( -1.5, 3)



13


a = 4b = 2

Focus 1 = ( 5.5, 3)

Focus 2 = ( -1.5, 3)

h= 2

k= 3

42 6-2-4-6

2

4

6

-2

-4

-6

8 10-8-10

8

-8

10

x

y


Center (2,3)



14

49x + 36y +392x - 360y - 80 = 0 2 2

49x +392x + 36y -360y - 80 = 02 2

49x +49(8)x + 36y -36(10)y -80 = 02 2

49 x + 8x + + 36 y -10y + -80 = 49 + 362 282

2 82

210 2

2 10 2

2

49 x + 8x + + 36 y -10 y + - 80 = 49 + 362 2 (4)2

(4)2 (5)

2(5)

2

16 (16)25 (25)49 x + 8x + + 36 y - 10y + - 80 = 49 + 362 2

49(x+4) + 36(y-5) - 80 = 784+9002 2

49(x+4) + 36(y-5) - 80 = 16842 2

+80 +80

49(x+4) + 36(y-5) = 17642 2

1764 1764

We know that 49x + 36y +392x -360y -80 = 0 is an ellipse. Put it in the standard form find a, b, c and the foci and graph it.

2 2

(x – h) (y – k)2 2= 1

b2 a2+



15

49(x+4) 36(y-5)2 2

=11764 1764

+

49(x+4) 36(y-5)2 2

=11764 176449

49 36

36

+

(x+4) (y-5)2 2

=136 49

+

(x-(-4)) (y-(+5))2 2

=136 49

+h= -4

k= 5

a = 492

b = 362

a = b + c2 2 2

-b -b2 2

c = a - b2 22

c = 49-362

c = 132

a = 492

b =362

a = 7

b = 6

Focus 1 = ( h, k + c)

Focus 2 = ( h, k - c)

Focus 1 = ( -4, 5 + ) 13

13Focus 2 = (-4, 5 - )

Focus 1 = ( -4, 5 + 3.6)

Focus 1 = ( -4, 8.6)

Focus 2 = ( -4, 5 – 3.6)Focus 2 = ( -4, 1.4)

49(x+4) + 36(y-5) = 17642 2

1764 1764

(x – h) (y – k)2 2= 1

b2 a2+

c = 13

c 3.6

49 > 36 So, this ellipse is vertical.

We know that 16x + 64y -64x -384y + 384 = 0 is an ellipse. Put it in the standard form find a, b, c and the foci and graph it.

2 2

(x – h) (y – k)2 2= 1

b2 a2+



16


a = 7b = 6

Focus 1 = (-4, 8.6)

Focus 2 = (-4, 1.4)

h= -4

k= 5


Center (-4,5) y

42 6-2-4-6

2

4

6

8 10-8-10

8

10

x

12

14

16

18



Documents

1 DEFINITION OF A ELLIPSE ELLIPSES PROBLEM 4 PROBLEM 1 PROBLEM 3 PROBLEM 2 STANDARD FORMULAS FOR ELLIPSES PROBLEM 5 Standards 4, 9, 16, 17 END SHOW PRESENTATION