Math Problem Solver

The graph of the equation \({x}^{2} - o \cdot .25 x - 0.125 = 0\) is a hyperbola.

The coordinates of its foci are: \(\left(- \frac{\sqrt{\frac{1}{4 + \sqrt{17}} - \frac{1}{4 - \sqrt{17}}}}{\sqrt{8 \sqrt{17} + 34}}, \frac{4 \sqrt{\frac{1}{4 + \sqrt{17}} - \frac{1}{4 - \sqrt{17}}}}{\sqrt{8 \sqrt{17} + 34}} + \frac{\sqrt{17} \sqrt{\frac{1}{4 + \sqrt{17}} - \frac{1}{4 - \sqrt{17}}}}{\sqrt{8 \sqrt{17} + 34}}\right)\) and \(\left(\frac{\sqrt{\frac{1}{4 + \sqrt{17}} - \frac{1}{4 - \sqrt{17}}}}{\sqrt{8 \sqrt{17} + 34}}, - \frac{\sqrt{17} \sqrt{\frac{1}{4 + \sqrt{17}} - \frac{1}{4 - \sqrt{17}}}}{\sqrt{8 \sqrt{17} + 34}} - \frac{4 \sqrt{\frac{1}{4 + \sqrt{17}} - \frac{1}{4 - \sqrt{17}}}}{\sqrt{8 \sqrt{17} + 34}}\right)\).

The coordinates of its vertices are: \(\left(- \frac{1}{\sqrt{4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}}, \frac{4}{\sqrt{4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}} + \frac{\sqrt{17}}{\sqrt{4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}}\right)\) and \(\left(\frac{1}{\sqrt{4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}}, - \frac{\sqrt{17}}{\sqrt{4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}} - \frac{4}{\sqrt{4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}}\right)\).

The asymptotes have equations: \(o \left(- \frac{2}{\sqrt{-4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}} + \frac{8}{\sqrt{4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}} + \frac{2 \sqrt{17}}{\sqrt{4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}}\right) + x \left(\frac{2}{\sqrt{4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}} + \frac{8}{\sqrt{-4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}} + \frac{2 \sqrt{17}}{\sqrt{-4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}}\right) = 0\) and \(o \left(- \frac{2}{\sqrt{-4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}} - \frac{2 \sqrt{17}}{\sqrt{4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}} - \frac{8}{\sqrt{4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}}\right) + x \left(- \frac{2}{\sqrt{4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}} + \frac{8}{\sqrt{-4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}} + \frac{2 \sqrt{17}}{\sqrt{-4 + \sqrt{17}} \sqrt{8 \sqrt{17} + 34}}\right) = 0\).