Find the equation of a circle with center at (-6, 4) and radius length 50. Then, give the domain and range of the relation. Give the answer in the simplest radical form, NOT IN DECIMAL FORM.
The equation of a circle with center at (h,k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
In this case, h = -6, k = 4, and r = 50. Substituting these values into the equation, we get:
(x + 6)^2 + (y - 4)^2 = 2500
The domain of this relation is all real numbers, since x can take on any value.
The range of this relation is also all real numbers, since y can take on any value.
To find the equation of a circle with center at (-6, 4) and radius length 50, we can use the standard form of the equation of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the coordinates of the center and r is the radius.
Given that the center is at (-6, 4) and the radius length is 50, the equation would be:
(x + 6)^2 + (y - 4)^2 = 50^2
Expanding this equation, we get:
(x + 6)^2 + (y - 4)^2 = 2500
Now let's find the domain and range of this relation.
The domain of a circle is the set of all possible x-values, which can be any real number since a circle extends indefinitely in the x-direction.
Therefore, the domain is (-∞, ∞).
The range of a circle is the set of all possible y-values. Since the center of the circle is at (h, k), the y-coordinate of the center gives us the vertical position of the circle. In this case, the center is at (−6, 4), so the range would be:
Range = k ± radius
Range = 4 ± 50
Therefore, the range is (-46, 54).