Question 251077
Here are the keys to solving this:<ul><li>The points where the two parts of a hyperbola are closest are the vertices.</li><li>The center of a hyperbola is halfway between the vertices.</li><li>The distance from the center of a hyperbola to one of its veritices is "a".</li><li>Since the center is halfway between the vertices and the distance from the center to a vertex is "a", then the distance between the two vertices is 2a.</li></ul>
So all we have to do to solve the problem is figure out what "a" is and then multiply it by 2.<br>
We can find "a" is we write the equation for the hyperbola in the form:
{{{(x-h)^2/a^2 - (y-k)^2/b^2 = 1}}}
So we want to transform {{{25x^2 - 81y^2 =   30625}}} into the form above. We can start by getting the 1 on the right by dividing both sides by 30625:
{{{25x^2/30625 - 81y^2/30625 = 1}}}
Next we can get rid of the coefficients of each fraction by multiplying the numerator and denominator of each fraction by the reciprocal of its coefficient:
{{{(25x^2/30625)((1/25)/(1/25)) - (81y^2/30625)((1/81)/(1/81)) = 1}}}
which gives us:
{{{x^2/(30625/25) - y^2/(30625/81) = 1}}}
which simplifies to
{{{x^2/1225 - y^2/(30625/81) = 1}}}
Since x = x-0 and y = y-0 we can rewrite this as:
{{{(x-0)^2/1225 - (y-0)^2/(30625/81) = 1}}}
And we have the desired form with {{{a^2 = 1225}}}. This makes a = 35 and the distance between the vertices, which is 2a, 70.