To find the center and radius of the circle given by the equation [tex]\((x - 7)^2 + (y + 4)^2 = 49\)[/tex], let's follow these detailed steps:
1. Identify the general form of the circle's equation:
The equation of a circle in standard form is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
2. Compare the given equation with the general form:
The given equation is [tex]\((x - 7)^2 + (y + 4)^2 = 49\)[/tex].
3. Extract the center coordinates from the equation:
- The term [tex]\((x - 7)^2\)[/tex] indicates that [tex]\(h = 7\)[/tex].
- The term [tex]\((y + 4)^2\)[/tex] indicates that [tex]\(k = -4\)[/tex] (because [tex]\(y + 4\)[/tex] is equivalent to [tex]\(y - (-4)\)[/tex]).
Therefore, the center [tex]\((h, k)\)[/tex] of the circle is [tex]\((7, -4)\)[/tex].
4. Determine the radius:
- The right-hand side of the equation [tex]\((x - 7)^2 + (y + 4)^2 = 49\)[/tex] is equal to [tex]\(49\)[/tex].
- This [tex]\(49\)[/tex] represents [tex]\(r^2\)[/tex], so to find the radius [tex]\(r\)[/tex], we take the square root of [tex]\(49\)[/tex].
Therefore, the radius [tex]\(r\)[/tex] is [tex]\(7\)[/tex].
So, the center of the circle is [tex]\((7, -4)\)[/tex] and the radius is [tex]\(7\)[/tex].
