Please write the equation of the conic section given the following information?: A hyperbola with vertices #(0,-6)# and #(0,6)# and asymptotes #y=3/4x# and #y=-3/4x#
1 Answer
Explanation:
The vertices of the hyperbola are told to be at (0, -6) and (0,6). Since these have different y-coordinates, but the same x-coordinate, we know that the vertices are located above and below each other, making this a vertical hyperbola.
The standard form of a vertical hyperbola is given by:
Center:
The center, by definition, is located in between the two vertices, or at (0,0) in this problem. This gives us:
Furthermore, the equations of the asymptotes for the hyperbola is given by:
Since we know that the asymptotes for this hyperbola are
If we keep in mind that the vertices are points on the graph, we know that (0,6) and (0,-6) both satisfy the equation. If we substitue (0,6) into our equation:
Since
Graph:
graph{(y^2/36 - x^2/64 - 1)(3/4x-y)(-3/4x-y) = 0 [-20, 20, -10.5, 10.5]}