Question

Дан эллипс 4х2 + 16у2 = 44. Найти его полуоси, координаты фокусов, эксцентриситет и уравнения директрис. Изобразить данный эллипс в системе координат

Answer 1

Дан эллипс 4х² + 16у² = 44. Из уравнения видно, что центр его находится в начале координат.

Выразим его уравнение в каноническом виде, разделив обе части на 44:  (4х²/44) + (16у/44) = 44/44.

(х²/11) + (у²/(11/4)) = 1.

Отсюда получаем длины полуосей: a = √11, b = √11/2.

Фокусы находятся на расстоянии "с" от центра.

с = √(a² - b²) = √(11 - (11/4) = √33/2.

Эксцентриситетом эллипса е = с/а = (√33/2)/√11 = √33/(2√11).

Прямые D1:x=−a/e и D2:x=a/e, перпендикулярные главной оси и проходящей на расстоянии a/e от центра, называются директрисами эллипса. D: x = +-(√11)/(√33/(2√11)) = +-(22/√33).

Answer 2

Ellipse Equation and Coefficients

The equation of the given ellipse is 4x^2 + 16y^2 = 44. This equation represents an ellipse centered at the origin (0,0) in the xy-plane.

To find the coefficients of the ellipse equation, we can rewrite it in standard form by dividing both sides by 44:

x^2/11 + y^2/2.75 = 1

From this equation, we can determine the coefficients of the ellipse:

- The coefficient of x^2 is 1/11. - The coefficient of y^2 is 1/2.75.

Semi-Axes of the Ellipse

The semi-axes of an ellipse are the lengths of the major and minor axes divided by 2. To find the semi-axes, we need to determine the square roots of the denominators of x^2 and y^2 in the standard form equation.

The semi-major axis (a) is the square root of the denominator of x^2, which is 11. So, a = sqrt(11).

The semi-minor axis (b) is the square root of the denominator of y^2, which is 2.75. So, b = sqrt(2.75).

Coordinates of the Foci

The coordinates of the foci can be found using the formula:

c = sqrt(a^2 - b^2)

where c is the distance from the center of the ellipse to each focus.

Substituting the values of a and b, we have:

c = sqrt(11 - 2.75) = sqrt(8.25)

The foci are located at (±c, 0). Therefore, the coordinates of the foci are:

F1: (-sqrt(8.25), 0) F2: (sqrt(8.25), 0)

Eccentricity

The eccentricity (e) of an ellipse is defined as the ratio of the distance between the foci to the length of the major axis. It can also be calculated using the formula:

e = c/a

Substituting the values of c and a, we have:

e = sqrt(8.25)/sqrt(11) = sqrt(8.25/11)

Therefore, the eccentricity of the ellipse is e = sqrt(8.25/11).

Equations of the Directrices

The equations of the directrices can be found using the formula:

x = ±a/e

Substituting the values of a and e, we have:

x = ±sqrt(11)/sqrt(8.25/11)

Simplifying further, we get:

x = ±sqrt(11/8.25)

Therefore, the equations of the directrices are:

x = ±sqrt(11/8.25)

Graph of the Ellipse

To graph the ellipse in the coordinate system, we can plot the center, foci, and vertices using the information we have obtained.

- Center: (0, 0) - Foci: F1: (-sqrt(8.25), 0) and F2: (sqrt(8.25), 0) - Vertices: (-sqrt(11), 0) and (sqrt(11), 0)

We can then sketch the ellipse using these points and the shape of the ellipse determined by the semi-axes.

Please note that the graph cannot be provided in this text-based format, but you can plot the points and sketch the ellipse on a graphing tool or software.