Question 1185283
<pre>
{{{36x^2  -  25y^2  -  72x + 50y  -  889}}}{{{""=""}}}{{{0}}}

Swap the middle two terms to get like lettered terms together:

{{{36x^2  -  72x -  25y^2 + 50y  -  889}}}{{{""=""}}}{{{0}}}

Factor 36 out of the two terms in x:  (Factor out 36, not 36x)

{{{36(x^2  -  2x) -  25y^2 + 50y  -  889}}}{{{""=""}}}{{{0}}}

Factor -25 out of the two terms in y:  (Factor out -25, not -25y)

{{{36(x^2  -  2x) -  25(y^2 + 2y)  -  889}}}{{{""=""}}}{{{0}}}

Add 889 to both sides:

{{{36(x^2  -  2x) -  25(y^2 + 2y)}}}{{{""=""}}}{{{889}}}

Complete the square inside the first parentheses:

1. To the side, multiply the coefficient of x, which is -2, by 1/2, getting -1.
2. Square this value, get +1
3. Add then subtract it "+1-1" inside the first parentheses

{{{36(x^2  -  2x+1-1) -  25(y^2 + 2y)}}}{{{""=""}}}{{{889}}} 

Complete the square inside the second parentheses:

1. To the side, multiply the coefficient of y, which is -2, by 1/2, getting -1.
2. Square this value, get +1
3. Add then subtract it "+1-1" inside the second parentheses

{{{36(x^2  -  2x+1-1) -  25(y^2 + 2y+1-1)}}}{{{""=""}}}{{{889}}}

Factor the first three terms inside the first parentheses using a large
parentheses with small parentheses inside:

{{{36( (x-1)(x-1)^""-1) -  25(y^2 + 2y+1-1)}}}{{{""=""}}}{{{889}}} 

Factor the first three terms inside the second parentheses using a large
parentheses with small parentheses inside:

{{{36((x-1)(x-1)^""-1) -  25((y-1)(y-1)^""-1)}}}{{{""=""}}}{{{889}}}

Write the products of binomials with themselves as the square of one
binomial:

{{{36((x-1)^2-1) -  25((y-1)^2-1)}}}{{{""=""}}}{{{889}}}

Remove the larger parentheses by distributing, leaving the smaller
parentheses intact:

{{{36(x-1)^2-36 -  25(y-1)^2+25}}}{{{""=""}}}{{{889}}}

Combine the two constant terms on the left

{{{36(x-1)^2-25(y-1)^2-11}}}{{{""=""}}}{{{889}}}

Add 11 to both sides

{{{36(x-1)^2-25(y-1)^2}}}{{{""=""}}}{{{900}}}

Divide each term on both sides by 900 to make the right side become 1:

{{{36(x-1)^2/900-25(y-1)^2/900}}}{{{""=""}}}{{{900/900}}}

Divide the numerators and denominators by the coefficients on top:

{{{(x-1)^2/25-(y-1)^2/36}}}{{{""=""}}}{{{1}}}

Compare this to

{{{(x-h)^2/a^2-(y-k)^2/36}}}{{{""=""}}}{{{1}}}

and we see that

h=1, k=1, a<sup>2</sup>=25, a=5, b<sup>2</sup>=36, b=6.

The center is (h,k) = (1,1).

We plot the center.  We draw the transverse axis which is a=5 units on both
sides of the center.  Then we draw the conjugate axis which is b=6 units vertically above and below the center.


{{{drawing(400,400,-9,11,-9,11,
graph(400,400,-9,11,-9,11),
green(line(1,-5,1,7),line(-4,1,6,1)) )}}}

We draw the defining rectangle so that the two axes bisect it:


{{{drawing(400,400,-9,11,-9,11,
graph(400,400,-9,11,-9,11),
green(line(1,-5,1,7),line(-4,1,6,1),line(-4,-5,-4,7),

line(-4,-5,6,-5),line(6,-5,6,7),line(6,7,-4,7)) )}}}

We draw and extend the two diagonals of the defining rectangle, which are the
asymptotes of the hyperbola:


{{{drawing(400,400,-9,11,-9,11,
graph(400,400,-9,11,-9,11),
green(line(1,-5,1,7),line(-4,1,6,1),line(-4,-5,-4,7),
line(-14,19,16,-17),
line(-4,-5,6,-5),line(6,-5,6,7),line(6,7,-4,7),
line(-14,-17,16,19)) )}}}

Now we sketch in the hyperbola:

{{{drawing(400,400,-9,11,-9,11,
graph(400,400,-9,11,-9,11,(1/5)(5-6sqrt(x^2-2x-24)) ),

graph(400,400,-9,11,-9,11,(1/5)(5+6sqrt(x^2-2x-24)) ),
green(line(1,-5,1,7),line(-4,1,6,1),line(-4,-5,-4,7),
line(-14,19,16,-17),
line(-4,-5,6,-5),line(6,-5,6,7),line(6,7,-4,7),
line(-14,-17,16,19)) )}}}

The center is (1,1)
The vertices are (-4,1) and (6,1)
To find the foci, we use the Pythagorean formula
to find their distance from the center

{{{c^2}}}{{{""=""}}}{{{a^2+b^2}}}
{{{c^2}}}{{{""=""}}}{{{5^2+6^2}}}
{{{c^2}}}{{{""=""}}}{{{25+36}}}
{{{c^2}}}{{{""=""}}}{{{61}}}
{{{c}}}{{{""=""}}}{{{sqrt(61)}}}

That is added to and subtracted from the x coordinate of the center to get
the coordinates of the foci. The foci are on the extension of the transverse
axes, shown below, and have the coordinates:

{{{ (matrix(1,3,1 +- sqrt(61),",",1))}}} 

{{{drawing(400,400,-9,11,-9,11,
graph(400,400,-9,11,-9,11,(1/5)(5-6sqrt(x^2-2x-24)) ),
circle(1-sqrt(61),1,.1), circle(1+sqrt(61),1,.1),
graph(400,400,-9,11,-9,11,(1/5)(5+6sqrt(x^2-2x-24)) ),
green(line(1,-5,1,7),line(-4,1,6,1),line(-4,-5,-4,7),
line(-14,19,16,-17),
line(-4,-5,6,-5),line(6,-5,6,7),line(6,7,-4,7),
line(-14,-17,16,19)) )}}}

Edwin</pre>