Example 1. Form the equation of a circle whose diameter is the segment cut off by the coordinate axes from the line $3 x-$ $-2 y+12=0$.
Solution. In Fig. 2.17, the line $3x - 2y + 12 = 0$ is depicted. It intersects the coordinate axes at points $A(-4,0)$ and $B(0,6)$.

![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-027.jpg?height=343&width=366&top_left_y=181&top_left_x=183)

Fig. 2.17

![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-027.jpg?height=359&width=503&top_left_y=167&top_left_x=669)

Fig. 2.18

1) The center of the circle is the point $M_{0}(x_{0}, y_{0})$ - the midpoint of segment $AB$. The coordinates of this point are determined by the formulas (see Chapter II, §1): $x_{0} = \frac{-4 + 0}{2} = -2$, $y_{0} = \frac{0 + 6}{2} = 3$, $M_{0}(-2,3)$.
2) The radius $R$ of the circle, equal to $A M_{0} \stackrel{2}{=} M_{0} B$, is calculated, for example, by the formula (see Chapter II, §1):

$$
R = \sqrt{(-4 + 2)^{2} + (0 - 3)^{2}} = \sqrt{4 + 9} = \sqrt{13}
$$

3) The canonical equation of the desired circle is $(x + 2)^{2} + (y - 3)^{2} = 13$.

Note. If the operations in the last equation are performed, we obtain the equation $x^{2} + 4x + y^{2} - 6y = 0$. This is called the general equation of the circle. It is an incomplete second-degree equation in terms of the variables $x$ and $y$.

## 4.2. Ellipse

An ellipse is the set of all points for which the sum of the distances to two fixed points, called foci, is a constant. (This constant is greater than the distance between the foci.)

If we assume that the foci of the ellipse are located at points $F_{1}(-c, 0)$ and $F_{2}(c, 0) (c > 0)$, and the constant is $2a$, then from its definition, we can derive the canonical equation of the ellipse:

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

Here, $a > 0$ is the major semi-axis, $b > 0$ is the minor semi-axis, $c$ is the focal distance, and $a^{2} = c^{2} + b^{2}$. The points $(a, 0)$ and $(-a, 0)$ are called the vertices of the ellipse.

The ellipse is illustrated in Fig. 2.18. Important characteristics of the ellipse include:

- Eccentricity $\varepsilon = \frac{c}{a} (0 < \varepsilon < 1)$; if $\varepsilon \approx 0$, the ellipse is nearly circular, i.e., close to a circle, and if $\varepsilon \approx 1$, the ellipse is flattened, close to the segment $[-a ; a]$;
- Directrices of the ellipse - lines with equations $x = \pm \frac{a}{\varepsilon}$;
- Distances from a point $M(x, y)$ on the ellipse to its foci ( $r_{1}$ to the left focus, $r_{2}$ to the right focus), calculated by the formulas:

$$
r_{1} = a + \varepsilon x, \quad r_{2} = a - \varepsilon x
$$

## Examples with Solutions