Question

Привести уравнение кривой второго порядка x^2 + 4y^2 + 4x - 8y - 8 = 0 к каноническому виду. Определить тип кривой, её параметры и сделть рисунок

Answer 1

Выделяем полные квадраты:
(x+2)² -4 +4*(y-1)² - 4 -8 =0
(x+2)² + 4*(y-1)² = 16

Разделим обе части уравнения на 16:
(x+2)² / 4² + (y-1)² / 2² = 1

Получили уравнение ЭЛЛИПСА 
Центр эллипса (-2; 1)
Полуоси:
a=4;  b=2

Чертим рисунок:(См скрин)

Answer 2

Converting the Equation to Canonical Form

To convert the equation of the second-order curve x^2 + 4y^2 + 4x - 8y - 8 = 0 to canonical form, we need to complete the square for both the x and y terms.

Let's start by rearranging the equation:

x^2 + 4x + 4y^2 - 8y - 8 = 0

Now, let's complete the square for the x terms. To do this, we add and subtract the square of half the coefficient of x:

x^2 + 4x + 4 + 4y^2 - 8y - 8 = 4

Simplifying further:

(x^2 + 4x + 4) + 4y^2 - 8y - 8 = 4

Now, let's complete the square for the y terms. To do this, we add and subtract the square of half the coefficient of y:

(x^2 + 4x + 4) + 4(y^2 - 2y + 1) - 8 - 4 = 4

Simplifying further:

(x^2 + 4x + 4) + 4(y^2 - 2y + 1) - 12 = 4

Now, let's simplify the equation:

(x + 2)^2 + 4(y - 1)^2 - 12 = 4

(x + 2)^2 + 4(y - 1)^2 = 16

Dividing both sides of the equation by 16, we get:

((x + 2)^2)/16 + ((y - 1)^2)/4 = 1

Type and Parameters of the Curve

The equation ((x + 2)^2)/16 + ((y - 1)^2)/4 = 1 represents an ellipse in the canonical form.

The center of the ellipse is at the point (-2, 1). The major axis is parallel to the x-axis and has a length of 8 units, while the minor axis is parallel to the y-axis and has a length of 4 units.

The semi-major axis is equal to the square root of the denominator of the x-term, which is 4. Therefore, the semi-major axis is 2 units.

The semi-minor axis is equal to the square root of the denominator of the y-term, which is 2. Therefore, the semi-minor axis is √2 units.

Graph of the Ellipse

To visualize the ellipse, we can plot the center (-2, 1) and draw the ellipse using the semi-major and semi-minor axes.

Here is a rough sketch of the ellipse:

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