Question 1082532
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<pre>
The literal translation is this equation

{{{((y+2)/(x-3))*((y-1)/(x+2))}}} = -6.


You can transform it:

(y+2)*(y-1) = -6*(x-3)*(x+2),

{{{y^2 + y -2}}} = {{{-6x^2 + 6x + 6}}},

{{{6x^2 - 6x + y^2 + y}}} = {{{8}}},

{{{6(x^2 - x)}}} + {{{(y^2 + y)}}} = {{{8}}},

{{{6*(x^2 - 2*(x/2) + (1/4)) - (6/4))}}} + {{{(y^2 + 2*(y/2) + (1/4)) - (1/4)}}} = {{{8}}},

{{{6(x-(1/2))^2}}} + {{{(y+(1/2))^2}}} = {{{8 + 6/4 + 1/4}}},

{{{6(x-0.5)^2}}} + {{{(y+0.5)^2}}} = {{{9.75}}},

{{{(x-0.5)^2/((9.75/6))}}} + {{{(y+0.5)^2/9.75}}} = {{{1}}},

{{{(x-0.5)^2/(sqrt(9.75/6))^2)}}} + {{{(y+0.5)^2/((sqrt(9.75))^2)}}} = {{{1}}},


You got the standard equation of the ellipse. 


Its center is at (0.5,-0.5) .

The major axis is vertical, while the minor axis is horizontal.

The major semi-axis has the length {{{sqrt(9.75)}}}.

The minor semi-axis has the length {{{sqrt(9.75/6)}}}.
</pre>

The solution is completed.



Regarding transformations from the general form equation to the standard form and identifying the ellipse elements 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/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/Identify-vertices-co-vertices-foci-of-the-ellipse-given-by-an-equation.lesson>Identify elements of an ellipse given by its general equation</A>

in this site.



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>".