246. Write the equation of an equilateral hyperbola, given that it passes through the point ( $1 ; 4$ ). Find the coordinates of the vertices of the hyperbola and the distance between them. Sketch the hyperbola.
Solution. According to the problem, $1 \cdot 4 = m$ or $m = 4$, hence the required equation of the hyperbola is $xy = 4$. Since $m = 4 > 0$, the branches of the hyperbola are located in the I and III quadrants. The vertices of the hyperbola are $A(2; 2)$ and $A_{1}(-2; -2)$.

![](https://cdn.mathpix.com/cropped/2024_05_22_1b9a1c1720bae4af5bc5g-053.jpg?height=367&width=429&top_left_y=468&top_left_x=659)

Fig. 30

The distance between the vertices of the hyperbola

$$
d = \sqrt{(2 + 2)^2 + (2 + 2)^2} = 4\sqrt{2} \approx 5.64
$$

First, we will construct the branch of the hyperbola in the I quadrant. We will create a table of positive values of $x, y$:

| $x$ | $\frac{4}{5}$ | 1 | $\frac{4}{3}$ | 2 | 3 | 4 | 5 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $y$ | 5 | 4 | 3 | 2 | $\frac{4}{3}$ | 1 | $\frac{4}{5}$ |

We will plot the points $B, C, D, A, E, F, K$, corresponding to the pairs of numbers in the table, and connect them with a smooth curve (Fig. 30).

To construct the branch of the hyperbola in the III quadrant, we can use the table we have created, taking the values of $x$ and $y$ with a negative sign, or plot a series of points ($E_{1}, D_{1}$, etc.), symmetric to arbitrary points of the constructed branch of the curve with respect to the origin, and connect them with a smooth curve.