SOLUTION: Two condominium apartment buildings that face each other have a front door awning that is shaped like a hyperbola. The equation of the hyperbola is 25x^2

SOLUTION: Two condominium apartment buildings that face each other have a front door awning that is shaped like a hyperbola. The equation of the hyperbola is 25x^2 - 81y^2 = 30,625. Find

Question 251077: Two condominium apartment buildings that face each other have a front door awning that is shaped like a hyperbola. The equation of the hyperbola is 25x^2 - 81y^2 = 30,625. Find the distance in meters between the two front door awnings at their closest point.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Here are the keys to solving this:

The points where the two parts of a hyperbola are closest are the vertices.
The center of a hyperbola is halfway between the vertices.
The distance from the center of a hyperbola to one of its veritices is "a".
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.

So all we have to do to solve the problem is figure out what "a" is and then multiply it by 2.

We can find "a" is we write the equation for the hyperbola in the form:

So we want to transform into the form above. We can start by getting the 1 on the right by dividing both sides by 30625:

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:

which gives us:

which simplifies to

Since x = x-0 and y = y-0 we can rewrite this as:

And we have the desired form with . This makes a = 35 and the distance between the vertices, which is 2a, 70.
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