Question

give the center and radius of the circle described by the equation and graph the equation. use the graph to identify the domain and range.
$(x + 5)^2+(y - 3)^2=4$
the center is \\( \square \\).
(type an ordered pair. simplify your answer.)
the radius is \\( \square \\).
(type an integer or a simplified fraction.)
graph the circle.
click to
enlarge
graph
express the domain of the relation in interval notation.
express the range of the relation in interval notation.

Explanation:

Step1: Recall the standard form of a circle equation

The standard form of a circle equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius.
For the given equation \((x + 5)^2+(y - 3)^2=4\), we can rewrite it as \((x-(- 5))^2+(y - 3)^2=2^2\).

Step2: Identify the center

Comparing with the standard form \((x - h)^2+(y - k)^2=r^2\), when \(h=-5\) and \(k = 3\), the center \((h,k)\) is \((-5,3)\).

Step3: Identify the radius

Comparing with the standard form \((x - h)^2+(y - k)^2=r^2\), when \(r^2 = 4\), then \(r=\sqrt{4}=2\).

Step4: Find the domain

The domain of a circle \((x - h)^2+(y - k)^2=r^2\) is given by \(h-r\leqslant x\leqslant h + r\).
Substituting \(h=-5\) and \(r = 2\), we get \(-5-2\leqslant x\leqslant-5 + 2\), which simplifies to \([-7,-3]\).

Step5: Find the range

The range of a circle \((x - h)^2+(y - k)^2=r^2\) is given by \(k-r\leqslant y\leqslant k + r\).
Substituting \(k = 3\) and \(r=2\), we get \(3-2\leqslant y\leqslant3 + 2\), which simplifies to \([1,5]\).

Answer:

The center is \((-5,3)\). The radius is \(2\). The domain is \([-7,-3]\). The range is \([1,5]\).