Question

Установите какую кривую определяет данное уровнение x^2+y^2-6x+10y+18=0

Answer 1

X²+y²-6x+10y+18=0     иксики к иксикам, а игреки - к игрекам, число вправо
x²-6x+y²+10y=-18     сгруппируем
(x²-6x)+(y²+10y)=-18      в каждой скобочке не хватает третьего слагаемого
(x²-6x+9)+(y²+10y+25)=-18+9+25     квадраты свернём, всё упростим
(х-3)²+(у+5)²=16 - это уравнение окружности с центром в точке (3;-5) и радиусом 4.

Answer 2

Equation Analysis:

The given equation is x^2 + y^2 - 6x + 10y + 18 = 0.

This equation represents a circle in the Cartesian coordinate system. The general form of the equation of a circle with center (h, k) and radius r is given by: x^2 + y^2 - 2hx - 2ky + h^2 + k^2 - r^2 = 0.

Comparing this general form with the given equation, we can determine the center and radius of the circle.

Center and Radius Calculation:

The center of the circle can be found by completing the square for x and y terms. The completed square form for x and y is given by: x^2 - 2hx + h^2 = (x - h)^2 and y^2 - 2ky + k^2 = (y - k)^2.

By comparing the given equation with the completed square forms, we can determine the center and radius of the circle.

Solution:

The completed square forms for x and y are: - x^2 - 6x + 9 = (x - 3)^2 - y^2 + 10y + 25 = (y + 5)^2

Comparing these with the given equation, we can rewrite the equation as: (x - 3)^2 + (y + 5)^2 - 9 - 25 + 18 = 0 (x - 3)^2 + (y + 5)^2 = 16

So, the center of the circle is at (3, -5) and the radius is 4.

Therefore, the given equation represents a circle with center at (3, -5) and radius 4.