Identify an equation in standard form for a hyperbola with center (0, 0), vertex (0, 17), and focus (0, 19).

Question

Please show how you arrive at answer.

Answer 1

To write the equation of the hyperbola, let’s closer examine the coordinates of the vertex and focus of the given hyperbola. We see that x-coordinates for vertex and focus are zeroes. We can conclude that the transverse axis in our case is vertical. Because the center of hyperbola is in origin, the equation of the given hyperbola has a form

                                                                y² / a²  - x² / b² = 1

Here  ±a –  is the y- coordinates of the vertices of the parabola (or y-intercepts),

b  determines the asymptotes of the hyperbola in the equation y = ± (a/b)x.

Because the vertex has coordinates (0.17), we see that a = ± 17.

It is given to us that the focus has coordinates (0, 19), hence the y-coordinate of it c = 19 .

Now to find b we can use the Pythagorean theorem

                                                                                         c² = a² + b².

After the substitution of known values, we will obtain:

                                                                                         19² = 17² + b²

From here b² = 19² – 17² = 361 – 289 = 72                  So b = √72 = 6√2

Now, when we found values of a and b, we finally get an answer

                                                                                y² / 17² – x² / (6√2)² = 1