SOLUTION: Find an equation of the ellipse that has center (-5,5), a minor axis of length 4, and a vertex at (4,5).

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Question 1137062: Find an equation of the ellipse that has center (-5,5), a minor axis of length 4, and a vertex at (4,5).
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Find an equation of the ellipse that has :
center (h,k)=(-5,5)
a minor axis of length b=4+
a vertex at (4,5)
since vertices are a+units to either side of the center, then a is distance from -5 to 4 and it is a=9+units

%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1
%28x-%28-5%29%29%5E2%2F9%5E2%2B%28y-5%29%5E2%2F4%5E2=1
%28x%2B5%29%5E2%2F81%2B%28y-5%29%5E2%2F16=1





Answer by ikleyn(52797) About Me  (Show Source):
You can put this solution on YOUR website!
.
The center (-5,5) and the vertex (4,5) are on one horizontal  line y= 5.

The distance between the center and given vertex is 4 - (-5) = 9 units.

It means that this distance IS NOT a minor semi-axes (!) : it is the MAJOR semi-axis.



So, we established that the major semi-axis is horizontal parallel to x-axis; its length is 9. 


Once again: the major semi-axis is parallel to x- axis and has the length of 9.

Hence, the minor semi-axis is parallel to y-axis and has the length 4/2 = 2.



The ellipse is longer than tall.


The equation has the form


%28x-%28-5%29%29%5E2%2F9%5E2 + %28y-5%29%5E2%2F2%5E2 = 1,    or


%28x%2B5%29%5E2%2F9%5E2 + %28y-5%29%5E2%2F2%5E2 = 1.

Solved.

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To see the theory and a set of solved problems on this subject, look into the lessons in this site
    - Ellipse definition, canonical equation, characteristic points and elements

    - Standard equation of an ellipse
    - Identify elements of an ellipse given by its standard equation
    - Find the standard equation of an ellipse given by its elements

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic
"Conic sections: Ellipses. Definition, major elements and properties. Solved problems".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.


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Be aware :   the solution by  @MathLover1 is   W R O N G !