Problem 64
Question
Give an example of a circle's equation in standard form. Describe how to find the center and radius for this circle.
Step-by-Step Solution
Verified Answer
The standard form of the given circle's equation is \((x-3)^2 + (y+4)^2 = 25\). The center of the circle is (3,-4) and the radius is 5.
1Step 1: Provide an example of a circle's equation in standard form
Let's consider the equation \((x-3)^2 + (y+4)^2 = 25\) as an example of a circle's equation in standard form.
2Step 2: Identify the center of the circle
From the equation, note that the term with \(x\) is \((x-3)\) and the term with \(y\) is \((y+4)\). Comparing this with the standard form, we see that \(h=3\) and \(k=-4\). Thus, the center of the circle is \((3,-4)\).
3Step 3: Determine the radius of the circle
From the equation, the right side is equal to 25, which is the square of the radius. Comparing this with the standard form, we infer that \(r^2 = 25\). Taking the square root of both sides, we get \(r = 5\). Therefore, the radius of the circle is 5.
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