# SOLUTION: y^2-x^2/15=1 Want to know the focus,vertices and the asymptote

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 Click here to see ALL problems on Quadratic-relations-and-conic-sections Question 621020: y^2-x^2/15=1 Want to know the focus,vertices and the asymptoteAnswer by KMST(1874)   (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).