<prompt> Identify the hyperbola that passes through the given points.Points: 3,4, 2,-6  <response>: A hyperbolas standard form is given by x-h2a2 - y-k2b21 for horizontal hyperbolas or y-k2a2 - x-h2b21 for vertical hyperbolas. Where h and k are the coordinates of the center, while a and b are constant values associated with the lengths of the axes of the hyperbola. So to identify the hyperbola that passes through both 3,4 and 2,-6, we will plug the points into the general equation of a hyperbola then solve for its variables. Assuming a hyperbola in the form x-h2a2 - y-k2b21 is what we are looking for, we plug 3,4 and 2,-6 into the equation to get the following system of equations:3-h2a2 - 4-k2b212-h2a2 - -6-k2b21Simplifying the system of equations we get:3-h2a2 - 4-k2b22-h2a2 - -6-k2b23-h2 - 4-k2 2-h2 - -6-k29 - 6h h2 - 16 8k - k2 4 - 4h h2 - 36 - 12k - k2h2 - 6h - k2 8k - 7 h2 - 4h - k2 -12k - 32h -5k - 25This seems to suggest that we dont have enough information to find a unique solution. Since we only have two points from a non-degenerate conic, there are infinitely many hyperbolas that pass through both points. 