Question 1169058
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This is the definition of a hyperbola with foci at (3,0) and (-3,0); that means the center is at (0,0), and the major axis is horizontal.  The general equation is<br>
{{{x^2/a^2-y^2/b^2=1}}}<br>
In that equation, a is the semi-major axis (distance from the center to each vertex) and b is the semi-minor axis.<br>
a and b are related by the equation<br>
{{{c^2 = a^2-b^2}}}<br>
where c is the distance from the center to each focus.<br>
In this example, then, we know c=3.<br>
It is easy to find the two points on the x-axis that are on the graph and are therefore the vertices.  With a distance of 6 between the two vertices, and a difference of 4 between the distances of a point from the two vertices, the vertices are at (2,0) and (-2,0); that makes a=2 and a^2=4.<br>
Then using {{{c^2=a^2-b^2}}}, we find {{{b^2=5}}}.<br>
So the equation of the locus is<br>
{{{x^2/4-y^2/5=1}}}<br>