Question 681942
Find the equation of a hyberbola with transverse axis on the line y= -5, length of transverse axis = 6, conjugate axis on the line x=2. and length of conjugate axis=6. Expres the answer in the form Ax^2 + Cy^2 + Dx + Ey + F = 0.
<pre>
The equation of such a hyperbola, with the transverse axis horizontal, is

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

where (h,k) is the center, a = {{{1/2}}} the length of the transverse axis,
and b = {{{1/2}}} the length of the conjugate axis.

We'll begin by drawing the horizontal line y = -5 and the vertical line x = 2.

{{{drawing(400,400,-8,12,-13,7, graph(400,400,-8,12,-13,7),green(line(-20,-5,20,-5),line(2,-20,2,20)) )}}}

The transverse axis and the conjugate axis intersect at the center of the
hyperbola which is (2,-5).  Now we'll leave just the transverse axis and the
conjugate axis, which are given as 6 units each, and we'll and erase the rest
of those green lines:

{{{drawing(400,400,-8,12,-13,7, graph(400,400,-8,12,-13,7),green(line(-1,-5,5,-5),line(2,-2,2,-8)) )}}}and draw the defining rectangle:{{{drawing(400,400,-8,12,-13,7, graph(400,400,-8,12,-13,7),green(line(-1,-5,5,-5),line(2,-2,2,-8)),rectangle(-1,-8,5,-2) )}}}

Now we can sketch in the asymptotes and the hyperbola:

{{{drawing(400,400,-8,12,-13,7, graph(400,400,-8,12,-13,7),green(line(-1,-5,5,-5),line(2,-2,2,-8)),rectangle(-1,-8,5,-2),line(-20,-27,30,23), line(30,-33,-31,28),graph(400,400,-8,12,-13,7,-5-sqrt((x-2)^2-9)),graph(400,400,-8,12,-13,7,-5+sqrt((x-2)^2-9)) 


 )}}}

We can write the equation of the hyperbola,

{{{(x-h)^2/z^2}}}{{{""-""}}}{{{(y-k)^2/b^2}}}{{{""=""}}}{{{1}}}

where (h,k) is the center (2,-5), a = {{{1/2}}} the length of the transverse
axis = {{{1/2}}}{{{""*""}}}{{{6}}} = 3
and b = {{{1/2}}} the length of the conjugate axis, also = 3

{{{(x-2)^2/3^2}}}{{{""-""}}}{{{(y+5)^2/3^2}}}{{{""=""}}}{{{1}}}

{{{(x-2)^2/9}}}{{{""-""}}}{{{(y+5)^2/9}}}{{{""=""}}}{{{1}}}

That is the equation in STANDARD, but the problem asks for it in the 
GENERAL form Ax² + Cy² + Dx + Ey + F = 0, so

{{{(x-2)^2/9}}}{{{""-""}}}{{{(y+5)^2/9}}}{{{""=""}}}{{{1}}}

Clear of fractions by multiplying through by 5

(x - 2)² - (y + 5)² = 9

x² - 4x + 4 - (y² + 10y + 25) = 9

x² - 4x + 4 - y² - 10y - 25 = 9

    x² - 4x - 21 - y² - 10y = 9

    x² - 4x - 30 - y² - 10y = 0

Rearrange the terms in the form Ax² + Cy² + Dx + Ey + F = 0

    x² - y² - 4x - 10y - 30 = 0


Edwin</pre>