Graphing ellipses

Question

Find the vertices, foci, center, lengths of major and minor axes, for each ellipse with the equation given below. Also graph each ellipse

 

x^2/16+y^2/36=1

 

4x^2+25y^2=100

 

(x+3)^2/5 + (y-1)^2/4=1 

Answer 1

Hi Lizzy,

 

Challenging problems for sure !!

 

Let's begin by considering the anatomy (parts) of an ellipse, according to its general equation:

    (x-h)²/a² + (y-k)²/b² = 1 (general equation)

    Center = (h,k)

    vertices:  (h +/- a,k); (h,k +/- b)

    foci (a > b):  (h +/- sqrt(a² - b²),k);  x major axis 

    foci (b > a):  (h,k +/- sqrt(b² - a²));  y major axis

 

Now we can apply the above info to each of our problems below:  

 

Problem #1:

 

x²/16 + y²/36 = 1

   Center (h,k) = (0,0)

   vertices:  (0 +/- 4,0); (0, +/- 6)

    foci:  (0,0 +/- sqrt(36-16)) = (0,0 +/- 4.47) 

 

Below is the link to our graph:

 

https://dl.dropbox.com/s/8ckwyv649zsujde/Graph_of_Ellipse_Problem_1.png?raw=1

 

Graph of ellipse is in blue with its center and four vertices marked in red and its two foci marked in green.

 

Problem #2:

 

4x² + 25y² = 100

 

To get in standard form, we will divide each term on both sides by 100, so that right side is 1:

 

x²/25 + y²/4 = 1 

    Center (h,k) = (0,0)

    vertices: (0 +/- 5,0); (0, +/- 2)
    foci: (0 +/- sqrt(25-4),0) = (0 +/- 4.58,0)

Below is the link to our graph:

https://dl.dropbox.com/s/gnnmcguh366lrb3/Graph_of_Ellipse_Problem_2.png?raw=1

 

Graph of ellipse is in blue with its center and four vertices marked in red and its  two foci marked in green. 

 

Problem #3:

(x²+3)/5 + (y²-1)/4 = 1
    Center (h,k) = (-3,1)
    vertices: (-3 +/- sqrt(5),1) = (-3 +/- 2,24,1);

                 (-3,1 +/- 2)
    foci: (-3 +/- sqrt(5-4),1) = (-3 +/- 1,1)

Below is the link to our graph:

https://dl.dropbox.com/s/mohflwxjpvt80fi/Graph_of_Ellipse_Problem_3.png?raw=1

Graph of ellipse is in blue with its center and four vertices marked in red and its two foci marked in green.

 

Thanks for submitting these problems and glad to help.

 

Take heart, Jordan (Romans 5:8)

Answer 2

Equation 1: x²/16 + y²/36 = 1

 

This is x²/4² + y²/6² = 1

 

Minor axis: 2a = 8 (horizontal)

 

Major axis: 2b = 12 (vertical)

 

Center: (h,k) = (0,0)

 

Vertices: (h,k±b) = (0,±6)

 

Foci: (h,k±√(b²-a²)) = (0,±√20) = (0,±2√5)

 

Equation 2: 4x² + 25y² = 100

 

This is x²/25 + y²/4 = 1 or x²/5² + y²/2² = 1

 

Minor axis: 2b = 4 (vertical)

Major axis: 2a = 10 (horizontal)

Center: (h,k) = (0,0)

Vertices: (h±a,k) = (±5,0)

Foci: (h±√(a²-b²),k) = (±√21,0)

 

Equation 3: (x+3)²/5 + (y-1)²/4 = 1

 

This is (x+3)²/(√5)² + (y-1)²/2² = 1

Minor axis: 2b = 4 (vertical)

Major axis: 2a =  2√5 (horizontal)

Center: (h,k) = (-3,1)

Vertices: (h±a,k) = (-3+√5,1) and (-3-√5,1)

Foci: (h±√(a²-b²),k) = (3±√1,1) = (4,1) and (2,1)