Part 3: Write the equation of and graph an ellipse.
Given the foci and vertices of an ellipse, complete the following steps.
Vertices at (4, -6 +-9)
Foci at (4, -6 +-5sqrt2)

a) What is the center of the ellipse? Explain how to determine this from the given points.

b) Write the standard equation of the ellipse. (3 points

c) What are the lengths of the major and minor axes? (2 points)
Major Axis:
Minor Axis:

d) Sketch a graph of the ellipse and label the center, foci, and the major and minor axes. (4 points)

Answer:

Step-by-step explanation:

The center is halfway between vertices, at (4, -6).

It is also halfway between foci.

:::::

The vertices are vertically aligned, so the parabola is vertical.

General equation for a vertical ellipse:

 (y-k)²/a² + (x-h)²/b² = 1

with

 center (h,k)

 vertices (h,k±a)

 co-vertices (h±b,k)

 foci (h,k±c), c² = a²-b²

Apply your data and solve for h, k, a, and b.

center (h,k) = (4, -6)

h = 4

k = -6

vertices (4,-6±a) = (4,-6±9)

a = 9

foci (4,-6±c) = (4,-6±5√2)

b² = a² - c² = 9² - (5√2)² = 31

b = √31

The equation becomes

 (y+6)²/81 + (x-4)²/31 = 1

:::::

length of major axis = 2a = 18

length of minor axis = 2b = 2√31

The various parts of the question needs to be answered.

Center: (4,-6)

Standard equation: [tex]\dfrac{(x-4)^2}{\sqrt{31}^2}+\dfrac{(y+6)^2}{9^2}=1[/tex]

Length of major axis: 18 units

Length of minor axis: [tex]2\sqrt{31}\ \text{units}[/tex]

Graph is attached.

The Vertices are of the form

[tex](h,k\pm a)=(4,-6\pm 9)[/tex]

This means that the ellipse is symmetrical to the y axis.

The center of an ellipse is given by [tex](h,k)[/tex]

So, here the center is [tex](4,-6)[/tex]

[tex]a=\pm 9[/tex]

The foci are of the form

[tex](h,k\pm c)=(4,-6\pm 5\sqrt{2})[/tex]

So, [tex]c=\pm 5\sqrt{2}[/tex]

[tex]b=\sqrt{a^2-c^2}\\\Rightarrow b=\sqrt{9^2-(5\sqrt{2})^2}\\\Rightarrow b=\pm\sqrt{31}[/tex]

The standard equation for major axis being parallel to y axis is given by

[tex]\dfrac{(x-h)^2}{b^2}+\dfrac{(y-k)^2}{a^2}=1\\\Rightarrow \dfrac{(x-4)^2}{\sqrt{31}^2}+\dfrac{(y+6)^2}{9^2}=1[/tex]

Length of major axis is

[tex]2a=2\times 9\\ =18\ \text{units}[/tex]

Length of minor axis is

[tex]2b=2\sqrt{31}\ \text{units}[/tex]

Learn more about ellipse:

https://brainly.com/question/12205108

How do you simplify 1 1/3

An investor has an account with stock from two different companies. Last year, her stock in Company A was worth $1300 and her stock in Company B was worth $4820. The stock in Company A has decreased 22% since last year and the stock in Company B has decreased 25%. What was the total percentage decrease in the investor's stock account? Round your answer to the nearest tenth (if necessary).

Which equation has no real solutions?

) ​​−2​x2=−168

) 4​x2+ 76 = 32

) 25​x2= 1

) 43 −​x2= 12

8 1/16-5 1/2 I need help with this.pls

Solve for x.
(Round to the nearest hundredth)

find the value of b in rectangle

f(x) = 4x - 9 step by step and put the right anwser so I can read it

Pls help me with this problem. Also the answer has to be in terms of pi

You are buying a new internet service. The service costs $75 per month. In the equation below, d is the
total number of dollars that you pay for your service, and t is the time (in months) that you keep the
service.
The relationship between these two variables can be expressed by the following equation:
d = 757
Identify the dependent and independent variables.
Help!!!

The equation x = 1 is an example of linear but not function. Why?

6. How many of the integers between 20 and 40 are
odd?
b)11
d)12
c12​

how do you solve this?

Michael had 3 times as many mangoes as Junior. After eating 8 mangoes. Michael had
twice as many mangoes as Junior. What is the total number of mangoes both boys had at
the beginning?​

Please I need help it’s an emergency

Can someone please tell me if this is right or what the right answer is pleaseeeeee

WILL GIVE BRAINLYIST

A cook needs 12 pounds of flour. He wants to spend the least amount of money. If a 2-pound bag costs $1.59 and a 5-pound bag costs $2.89, how many bags of each type of flour should he buy? What will be the total cost?

PLS HELP I DONT KNOW THIS ONE ITS URGENT

What is the slope-intercept equation for the line below?
5
(4,3)
(0, 1)
5
O A. y=2x+1
B. y=-3x+1
O C. y=-2x+1
O D. y = 2x+1

What is an angle call measure between 90° and 180°

6 3/4"
What is 6 divided by 3/4

Can someone help me

I have no idea what to do

N - 5 x4 pls hellpppo

How do you find the slope to (3,12),(3,-17)

Help pls and thank you ..

Need help ASAP !!!
Solve for u

Which problem can be solved with this equation?
A. Jen has 4 of a meter of ribbon. She buys 6 more meters of ribbon. How many meters of ribbon does she have in all?
B. Jen has f of a meter of ribbon, she wants to give 6 friends equal-size pleces of the ribbon. How many meters of ribbon does each
friend get?
OC. Jen has 6 meters of ribbon. She cuts
soft / of
of the ribbon. How many meters of ribbon does she have left?
OD. Jen has 6 meters of ribbon, she wants to give
of it to her friend. How many meters of ribbon does she give her friend?

students recorded the total number of pages and chapters in each of several books on the scatterplot. Based on the sbcatterplot, which is the best prediction of the number of chapters in a book with 200 pages

PLEASE ANSWER FAST!!!!!