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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~


<pre>
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

{{{(x-2)^2/1^2}}} + {{{(y+4)^2/5^2}}} = 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  {{{sqrt(5^2-1^2)}}} = {{{sqrt(24)}}} = {{{2*sqrt(6)}}}.


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+{{{2*sqrt(6)}}})  and  (2,-4-{{{2*sqrt(6)}}}).


The plot is THIS:


{{{drawing( 330, 330, -4.5, 8.5, -10.5, 2.5,
            circle(2, -4-2sqrt(6), 0.12), circle(2, -4, 0.12), circle(2, -4+2sqrt(6), 0.12),

   graph( 330, 330, -4.5, 8.5, -10.5, 2.5,
          -4 + 5*sqrt(1-((x-2)^2/1)),
          -4 - 5*sqrt(1-((x-2)^2/1)))
)}}}


&nbsp;Ellipse {{{(x-2)^2/1^2}}} + {{{(y+4)^2/5^2}}} = 1
</pre>


See the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Ellipse-definition--canonical-equation--characteristic-points-and-elements.lesson>Ellipse definition, canonical equation, characteristic points and elements</A> 


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Standard-equation-of-an-ellipse.lesson>Standard equation of an ellipse</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Identify-elements-of-an-ellipse-given-by-its-standard-eqn.lesson>Identify elements of an ellipse given by its standard equation</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Find-a-standard-equation-of-an-ellipse-given-by-its-elements.lesson>Find the standard equation of an ellipse given by its elements</A>


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/General-equation-of-an-ellipse.lesson>General equation of an ellipse</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Transform-general-eqn-of-an-ellipse-to-the-standard-form-by-completing-the-square.lesson>Transform a general equation of an ellipse to the standard form by completing the square</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Identify-vertices-co-vertices-foci-of-the-ellipse-given-by-an-equation.lesson>Identify elements of an ellipse given by its general equation</A>



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


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



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.