An architect designs two houses that are shaped and positioned like a part of the branches of the hyperbola whose equation is $225 y^{2}-1225 x^{2}=275,625$, where $x$ and $y$ are in yards. How far apart are the houses at their closest point?

Step 1 ：The given equation is of a hyperbola. The standard form of a hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\).

Step 2 ：The given equation can be rewritten in the standard form by dividing both sides by 275,625 and rearranging the terms.

Step 3 ：The distance between the houses at their closest point would be the distance between the vertices of the hyperbola, which is \(2a\) for a hyperbola of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(2b\) for a hyperbola of the form \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\).

Step 4 ：Given that \(a^2 = 225.0\) and \(b^2 = 1225.0\), we can calculate the distance as \(2a = 2\sqrt{a^2} = 2\sqrt{225} = 30\) or \(2b = 2\sqrt{b^2} = 2\sqrt{1225} = 70\).

Step 5 ：Final Answer: The houses are \(\boxed{70}\) yards apart at their closest point.