Question

an architect designs two houses that are shaped and positioned like a part of the branches of the hyperbola whose equation is 2025y² - 625x² = 1,265,625, where x and y are in yards. how far apart are the houses at their closest point? the houses are \boxed{} yards apart. (simplify your answer.)

Explanation:

Step1: Rewrite the hyperbola equation in standard form.

The standard form of a vertical hyperbola is \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1\). Given the equation \(2025y^{2}-625x^{2}=1265625\), divide each term by \(1265625\):

$$ \frac{2025y^{2}}{1265625}-\frac{625x^{2}}{1265625}=1 $$

Simplify the fractions: \(\frac{y^{2}}{\frac{1265625}{2025}}-\frac{x^{2}}{\frac{1265625}{625}} = 1\). Calculate \(\frac{1265625}{2025}=625\) and \(\frac{1265625}{625}=2025\). So the equation becomes \(\frac{y^{2}}{625}-\frac{x^{2}}{2025}=1\).

Step2: Identify the value of \(a\).

For the standard form \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1\), we have \(a^{2}=625\), so \(a = \sqrt{625}=25\).

Step3: Determine the closest distance between the two branches.

The closest distance between the two branches of a vertical hyperbola (opening up and down) is \(2a\) (the distance between the vertices). So \(2a=2\times25 = 50\).

Answer:

50