SOLUTION: Given the equation 25x^2+y^2-100x+8y+91=0 a. Write the equation in the standard form b. Find the coordinates of the center, foci, vertices, andd co-vertices c. Sketch the graph

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Question 1101234: Given the equation 25x^2+y^2-100x+8y+91=0
a. Write the equation in the standard form
b. Find the coordinates of the center, foci, vertices, andd co-vertices
c. Sketch the graph of the ellipse

Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.
Given the equation 25x^2+y^2-100x+8y+91=0
a. Write the equation in the standard form
b. Find the coordinates of the center, foci, vertices, andd co-vertices
c. Sketch the graph of the ellipse
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Technically speaking, you need to complete squares separately for x- and y- terms.


25x^2 + y^2 - 100x + 8y + 91 = 0     is equivalent to

(25x^2 - 100x) + (y^2 + 8y) = - 91   is equivalent to

25*(x^2 - 4x)  + (y^2 + 8y) = -91    is equivalent to

25*(x^2 - 4x + 4) + (y^2 + 8y + 16) = -91 + 100 + 16    is equivalent to

25*(x-2)^2 + (y+4)^2 = 25      (<<<---=== divide by 25 both sides ===--->>>)  is equivalent to

 +  = 1.


The last equation is the standard form equation of the ellipse 

   - centered at the point (2,-4)

   - with the major axis parallel to y-axis

   - taller than wide

   - with the major semi-axis of the length of 5 in vertical   direction and
          the minor semi-axis of the length of 1 in horizontal direction.


The linear eccentricity of the ellipse is   =  = .


The    vertices are  (2,-4+5) = (2,1)   and  (2,-4-5) = (2,-9).

The co-vertices are  (2+1,-4) = (3,-4)  and  (2-1,-4) = (1,-4).


The foci are  (2,-4+)  and  (2,-4-).


The plot is THIS:





 Ellipse  +  = 1


See the lessons
    - 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

    - General equation of an ellipse
    - Transform a general equation of an ellipse to the standard form by completing the square
    - Identify elements of an ellipse given by its general equation


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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