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Question 621020: y^2-x^2/15=1
Want to know the focus,vertices and the asymptote
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! For hyperbolas, we like the standard form of the equation, the one that shows a difference of squares equal to 1.
It may look like
 or  .
That form of the equation shows you all the numbers you need to know to figure out the he foci, vertices and the asymptotes. (All you need is the a and b numbers).
 is the equation (in standard form) of a hyperbola centered at the origin, so this is an easy problem.
We know this one is centered at the origin because there is just an  and a  , with nothing added or subtracted before squaring.
Because of that simplicity, it is easy to see that changing x to -x gives you the same equation, meaning that the graph is symmetrical with respect to the y-axis. The same can be said of changing y to -y, and the symmetry with respect to the x-axis.
For y=0 we would have a negative number equal to 1  and that cannot be.
So, we can see that the graph does not touch the x-axis, where y=0. (In fact the graph does not even want to get close to the x-axis)
On the other hand, y cannot be zero, but x can be zero.
When  , you see that  , meaning  or  , so the graph goes through the points (0,1) and (0,-1).
For all other points,  so  and  , meaning that all the other points are even farther away from the x-axis, where y=0.
The closest that the hyperbola comes to the x-axis is the points (0,1) and (0,-1) , which are the vertices.
As x (and y) grow larger in absolute value, and grow larger, and the graph gets closer to the asymptotes.
A little algebra transforms the equation into one that gives us the equations of the asymptotes:
-->  -->  -->  --> 
As grows larger,  grows smaller, and the graph gets closer to the graph for
 which is the graph for the lines
 <-->  and
 <-->  .
Those lines are the asymptotes.
Because teachers do not like to see square roots in denominators, we may have to write them as
 and  .
THE FOCI:
There foci are at a distance  from the center of the hyperbola, and the number is related to the numbers  and  in the standard form of the equation by a formula that can be derived using the Pythagorean theorem. It is
In this case, your and are 1 and 15, so
 -->  -->  .
The center was (0,0) (the origin).
The vertices were ((0,-1) and (0,1), on the y-axis.
The foci are on the same line, but at distance 4 from the center/origin, at
(0,-4) and (0,4).
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