Example 5. Write the polar equation $r=\frac{2}{4-\sin \varphi}$ in rectangular coordinates
Solution. First, rewrite the given equation in the form $4 r = r \sin \varphi + 2$ and use the formulas (substitutions) $r = \sqrt{x^{2} + y^{2}}$, $r \sin \varphi = y$. We get: $4 \sqrt{x^{2} + y^{2}} = y + 2 \ (y + 2 \geqslant 0)$. Next, by squaring this equality and after transformations and completing the square, we obtain: $16(x^{2} + y^{2}) = y^{2} + 4y + 4, \ 16x^{2} + 15y^{2} - 4y = 4, \ 16x^{2} + 15\left(y - \frac{2}{15}\right)^{2} = \frac{64}{15}, \ \frac{x^{2}}{\left(\frac{2}{\sqrt{15}}\right)^{2}} + \frac{\left(y - \frac{2}{15}\right)^{2}}{\left(\frac{8}{15}\right)^{2}} = 1$. We have obtained the canonical equation of an ellipse with the center at the point $\left(0, \frac{2}{15}\right)$ and semi-axes $a = \frac{2}{\sqrt{15}}$ and $b = \frac{8}{15}$.

## 4.3. Hyperbola

$1^{\circ}$. A hyperbola is defined as the set of all points (GMT) for which the absolute value of the difference of the distances to two fixed points, called foci, is a constant. (This constant is less than the distance between the foci.)

$2^{\circ}$. If the foci of the hyperbola are located at points $F_{1}(-c, 0)$, $F_{2}(c, 0), c>0$, and this constant is equal to $2a$, then such a hyperbola has the canonical equation

$$
\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1
$$

where $a^{2} + b^{2} = c^{2}$.

Here, $a$ is the real semi-axis, $b$ is the imaginary semi-axis $(a>0, b>0)$, and $c$ is the focal distance $(c>0)$ (see Fig. 2.21).

$3^{\circ}$. The lines with equations $y = \pm \frac{b}{a} x$ are called the asymptotes of the hyperbola. The quantity

![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-028.jpg?height=415&width=582&top_left_y=1151&top_left_x=1900)

Fig. 2.21 $\varepsilon = \frac{c}{a} (1 < \varepsilon < +\infty)$ is called the eccentricity of the hyperbola (for large $\varepsilon$, the branches of the hyperbola are wide and almost vertical, while for $\varepsilon \approx 1$, the branches of the hyperbola are narrow and the hyperbola approaches the $Ox$ axis).

The distances from a point $M(x, y)$ on the hyperbola to its foci ($r_{1}$ from the left focus, $r_{2}$ from the right focus) are: $r_{1} = |a + \varepsilon x|, \quad r_{2} = |a - \varepsilon x|$.

The lines with equations $x = \pm \frac{a}{\varepsilon}$ are called the directrices of the hyperbola.

## Examples with Solutions