Question 1087980
.
<pre>
The length of the real axis of the hyperbola is 6 - 2 = 4,
and the real axis is the horizontal line y = 0 coinciding with x-axis.


The real semi-axis length is a = {{{4/2}}} = 2.


The center of the hyperbola is the midpoint between (0,0) and (8,0).
So, the center is the point (4,0).


The foci distance is 8 - 0 = 8.
The half of the foci distance is {{{8/2}}} = 4.
It is the distance from the hyperbola center to the focus point, which is traditionally called "c".


The imaginary semi-axis "b" is  {{{b^2}}} = {{{c^2 - a^2}}} = {{{4^2 - 2^2}}} = 12.
Hence, b = {{{sqrt(12)}}}.


Thus the hyperbola standard form equation is

{{{(x-4)^2/a^2}}} - {{{(y-0)^2/(sqrt(12))^2}}} = 1,


or, which is the same,  

{{{(x-4)^2/4}}} - {{{y^2/12}}} = 1.


{{{drawing(330, 330, -2.5, 8.5, -4.5, 4.5,
           circle(4-4, 0, 0.15), circle(4+4, 0, 0.15), circle(4, 0, 0.15),
           circle(4-4, 0, 0.08), circle(4+4, 0, 0.08), circle(4, 0, 0.08),

     graph(330, 330, -2.5, 8.5, -4.5, 4.5,
           sqrt(12)*sqrt((x-4)^2/4-1),
          -sqrt(12)*sqrt((x-4)^2/4-1))
)}}}


Hyperbola {{{(x-4)^2/4}}} - {{{y^2/12}}} = 1  


The foci are (0,0) and (8,0).
</pre>

See the lessons

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

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

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

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


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/REVIEW-of-lessons-on-hyperbolas.lesson>OVERVIEW of lessons on hyperbolas</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: Hyperbolas. Definition, major elements and properties. Solved problems</U>".