Question 1086650
.
find the focus of x^2+4y^2+2x+24y++28=0
~~~~~~~~~~~~~~~~~~~~~~~~~~~


I don't know what "++" means; will assume that it is "+".


<pre>
{{{x^2+4y^2+2x+24y+28}}] = {{{0}}}  ===> (I will complete the squares for x-terms and y-terms separately)  ====>  

{{{x^2+4y^2+2x+24y}}} = {{{-28}}},

{{{(x^2 + 2x)}}} + {{{(4y^2 + 24y)}}} = {{{-28}}},

{{{(x^2 + 2x + 1)}}} + {{{((2y)^2 + 2*(2y*6) + 36)}}} = {{{-28 + 1 + 36}}},

{{{(x+1)^2}}} + {{{(2y+6)^2}}} = {{{9}}},

{{{(x+1)^2/3^2}}} + {{{(y+3)^2/(3/2)^2}}} = 1.



{{{drawing(330, 330, -4.5, 2.5, -5.5, 1.5,
           circle(-1-sqrt(6.75), -3, 0.06), circle(-1, -3, 0.06), 
           circle(-1+sqrt(6.75), -3, 0.06),

     graph( 330, 330, -4.5, 2.5, -5.5, 1.5,
          -3 + (3/2)*sqrt(1-(x+1)^2/9), -3 - (3/2)*sqrt(1-(x+1)^2/9))
)}}}


Ellipse {{{(x+1)^2/3^2}}} + {{{(y+3)^2/(3/2)^2}}} = 1



The center of the ellipse is at (x,y) = (-1,-3).


The ellipse has the horizontal major axis.


The major semi-axis is 3 units long.


The minor semi-axis is {{{3/2}}} = 1.5 units long.


The linear eccentricity is {{{sqrt(3^2 - 1.5^2)}}} = {{{sqrt(6.75)}}}.


The foci are at  ({{{-1-sqrt(6.75)}}},{{{-3}}})  and  ({{{-1+sqrt(6.75)}}},{{{-3}}}).
</pre>

For more details, 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/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>".