Precalculus Lesson: Ellipse - 04/01/24

Horizontal Major Axis - COPY DOWN!

• Vertices Coordinates:

• (h - a, k) and (h + a, k)

• Co-Vertices Coordinates:

• (h, k - b) and (h, k + b)

Conic Sections

Circles

Ellipses

Parabolas

Hyperbolas

Horizontal Ellipse has Horizontal al Major Axis

• Center: (h, k)

• Vertices Coordinates: (h - a, k) and (h + a, k)

• Co-Vertices Coordinates: (h, k - b) and (h, k+b)

• Foci coordinate (h-c, k) and (h+c, k)

• Equation of foci: c² = a² - b²

Vertical Ellipse has vertical Major Axis

• Center: (h, k)

• Vertices Coordinates: (h, k-a) and (h, k+a)

• Co-Vertices Coordinates: (h-b, k) and (h+b, k)

• Foci coordinate (h, k+c) and (h, k-c)

• Equation of foci: c² = a² - b²

What are the a and b used for? - COPY DOWN!

• Other than telling us if the graph is Horizontal or Vertical:

• 'a' values are used to find the Vertices points

• 'b' values are used to find the Co-Vertices points

Multiple Choice

The a² is always assigned to the:

• Larger Denominator

• Smaller Denominator

Multiple Choice

What is the equation to get the foci of the ellipse?

• $$c^2=a^2-b^2$$

• $$c^2=a^2+b^2$$

• $$c=\sqrt{a-b}$$

• $$c=\sqrt{a+b}$$

Multiple Choice

Find the standard equation of the given ellipse

Multiple Choice

What are the coordinates of the foci of the given ellipse?

• (-3 , 4) and (-3 , 0)

• (3 , -4) and (3 , 0)

• (-3 , -4) and (-3 , 0)

• (-3 , 4) and (3 , 0)

Multiple Choice

Using the Formulas for the Vertices, what would be the correct Vertices for this example from the previous slide?

• (-3, -2) and (1, -2)

• (-5, -5) and (0, -3)

• (3, 2) and (5, 2)

• (-5, -2) and (1, -2)

Multiple Choice

An ellipse centered at the origin has a vertical major axis length of 8 units and an horizontal minor axis length of 4. What is the equation of the ellipse?

• $$\frac{x^2}{4}+\frac{y^2}{16}=1$$  

• $$\frac{x^2}{16}+\frac{y^2}{4}=1$$  

• $$\frac{x^2}{16}+\frac{y^2}{64}=1$$  

• $$\frac{x^2}{64}+\frac{y^2}{16}=1$$  

Multiple Choice

An ellipse has vertices at (2, 2) and (2, 8) and co-vertices at (0, 5) and (4, 5). What is the equation of the ellipse?

• $$\frac{\left(x-2\right)^2}{4}+\frac{\left(y-5\right)^2}{9}=1$$  

• $$\frac{\left(x+2\right)^2}{4}+\frac{\left(y+5\right)^2}{9}=1$$  

• $$\frac{\left(x-2\right)^2}{9}+\frac{\left(y-5\right)^2}{4}=1$$  

• $$\frac{\left(x+2\right)^2}{9}+\frac{\left(y+5\right)^2}{4}=1$$  

Equations of Ellipses - COPY DOWN!!!!

• a² is always assigned to the largest denominator

• If the larger denominator is under the x, it is a Horizontal ellipse

• If the larger denominator is under the y, it is a Vertical ellipse

• Note that the Center of the Ellipse still uses the same (h,k) as the Circle did

Draw

DO IT RESPONSE CARD: Identify the conic and it's vertices

$$\left(x-3\right)^2+4\left(y+5\right)^2=144$$  

Multiple Choice

Identify the conic and it's vertices

$$\left(x-3\right)^2+4\left(y+5\right)^2=144$$  

• horizontal ellipse; V(15, -5) and (-9, 2)

• Horizontal V(-9, -5) and (-15, -5)

• vertical ellipse; V(-9, 15) and (5, -5)

• vertical ellipse; V(-2, 2) and (-2, 6)

• Horizontal V(-9, 15) and (5, -5)

Draw

DO IT ON RESPONSE CARD: Identify the conic and it's vertices

$$4\left(x+2\right)^2+\left(y-2\right)^2=36$$  

Multiple Choice

Identify the conic and it's vertices

$$4\left(x+2\right)^2+\left(y-2\right)^2=36$$  

• horizontal ellipse; V(-8, 2) and (4, 2)

• vertical ellipse; V(-2, -4) and (-2, 8)

• vertical ellipse; V(-2, 4) and (-2, -8)

• horizontal ellipse; V(-2, 2) and (-2, 6)

Draw

DO IT IN JOURNAL: Find the equation of the ellipse in standard form.

$$4x^2+y^2+8x\ -2y-11=0$$  

Multiple Choice

Find the equation of the ellipse in standard form.

$$4x^2+y^2+8x\ -2y-11=0$$  

• $$\frac{\left(x+1\right)^2}{2}+\frac{\left(y-1\right)^2}{8}=1$$  

• $$\frac{\left(x-1\right)^2}{16}+\frac{\left(y+1\right)^2}{4}=1$$  

• $$\frac{\left(x+1\right)^2}{4}+\frac{\left(y-1\right)^2}{16}=1$$  

• $$\frac{\left(x+1\right)^2}{4}+\frac{\left(y+1\right)^2}{2}=1$$  

Draw

DO IT ON WHITEBOARD: Name the conic section and find the center

25x² + 9y² + 50x - 36y - 164 = 0

Multiple Choice

Name the conic section and find the center

25x² + 9y² + 50x - 36y - 164 = 0

• vertical ellipse; C(-1, 2)

• horizontal ellipse; C(-1, 2)

• horizontal ellipse; C(-2, 4)

• vertical ellipse; C(1, -2)

Draw

Do it on whiteboard: An ellipse has vertices at (-3, 2) and (5, 2) and co-vertices at (1, -1) and (1, 5). What is the equation of the ellipse?

Multiple Choice

An ellipse has vertices at (-3, 2) and (5, 2) and co-vertices at (1, -1) and (1, 5). What is the equation of the ellipse?

• $$\frac{\left(x-1\right)^2}{16}+\frac{\left(y-2\right)^2}{9}=1$$  

• $$\frac{\left(x+1\right)^2}{16}+\frac{\left(y+2\right)^2}{9}=1$$  

• $$\frac{\left(x-1\right)^2}{9}+\frac{\left(y-2\right)^2}{16}=1$$  

• $$\frac{\left(x+1\right)^2}{9}+\frac{\left(y+2\right)^2}{16}=1$$  

Draw

DO IT ON WHITEBORAD: An ellipse has co-vertices at (-4, -3) and (-10, -3) and foci at (-7,  1) and (-7, -7). What is the equation of the ellipse?

Multiple Choice

An ellipse has co-vertices at (-4, -3) and (-10, -3) and foci at (-7,  1) and (-7, -7). What is the equation of the ellipse?

• $$\frac{\left(x+3\right)^2}{9}+\frac{\left(y+7\right)^2}{25}=1$$  

• $$\frac{\left(x-7\right)^2}{9}+\frac{\left(y-3\right)^2}{25}=1$$  

• $$\frac{\left(x+7\right)^2}{25}+\frac{\left(y+3\right)^2}{9}=1$$  

• $$\frac{\left(x+7\right)^2}{9}+\frac{\left(y+3\right)^2}{25}=1$$  

Draw

DO IT ON JOURNAL: An ellipse has the following set of characteristics:

Vertices ( 4, 3), (4, - 9)

Length of minor axis is 8

What is the center of this ellipse?

Multiple Choice

An ellipse has the following set of characteristics:

Vertices ( 4, 3), (4, - 9)

Length of minor axis is 8

What is the center of this ellipse?

• (4, 6)

• ( - 4, 3)

• (6, 4)

• (4, -3)

Draw

DO IT ON WHITEBORAD: Write an equation for the ellipse with each set of characteristics. Then answer the question.
Vertices ( -7, -3), (13, - 3)
foci ( - 5, -3 ) , (11 , -3)
what is the a and b value of the ellipse?

Multiple Choice

Write an equation for the ellipse with each set of characteristics. Then answer the question.
Vertices ( -7, -3), (13, - 3)
foci ( - 5, -3 ) , (11 , -3)
what is the a and b value of the ellipse?

• a = 10 , b = 6
• a = 6 , b = 10
• a = 36 , b = 100
• a = 100 , b = 36

Poll

How Are You Today?

• Awesome

• "I'm in math class, so...yeah"

• Okay

What's a Foci? - COPY DOWN!!

• Foci is the plural of Focus

• You'll remember that Parabolas had just one Focus

• Since Ellipses have two of them, they are called Foci, or F1 and F2 symbolically

Multiple Choice

What is the Term that we used for the plural of Focus?

• Focuses

• Foci

• Fungi

• Ford

Multiple Choice

How many Foci are there in an Ellipse

• 0

• 1

• 2

• 3

Multiple Choice

If the Larger Denominator is under the y, then the Ellipse will be:

• Vertically Positioned

• Horizontally Positioned

What are the a and b used for? - COPY DOWN!

• Other than telling us if the graph is Horizontal or Vertical:

• 'a' values are used to find the Vertices points

• 'b' values are used to find the Co-Vertices points

What are Vertices and Co-Vertices? COPY DOWN!!!

• Vertices are the two points on the Ellipse that are FARTHEST from the Center

• Co-Vertices are the two points on the Ellipse that are CLOSEST to the Center

Multiple Choice

Vertices points are _______ from the Center, while Co-Vertices points are _______ from/to the Center

• Closest, Farthest

• Farthest, Closest

Horizontal Major Axis - COPY DOWN!

• Vertices Coordinates:

• (h - a, k) and (h + a, k)

• Co-Vertices Coordinates:

• (h, k - b) and (h, k + b)

Example: COPY DOWN!!!

• Center is at (-2, -2) Remember, h and k are opposite from the equation

• a² would be assigned to the 9, which means a = 3

• By default, that means that b² would be assigned to the 4, which would lead to b = 2

• This would be a Horizontal Major Axis Ellipse, because the a² is under the x

Multiple Choice

Using the same example, using our formulas for the Co-Vertices, what are the correct coordinates for the Co-Vertices?

• (-2, 0) and (-2, -4)

• (-2, -2) nd (-2, 3)

• (-5, -2) and (1, -2)