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To find the center of a circle when you're given the equation in standard form, focus on the terms that involve variables squared. A circle's equation typically looks like this: \[(x-h)^2 + (y-k)^2 = r^2\] where

• \(h\)
• \(k\)

are the coordinates of the circle's center.The equation \[(x+6)^2 + y^2 = 121\] looks a bit different, but you can still spot the center.For the x-term, \((x+6)\),it can be rewritten as \((x - (-6))\). This tells us that the x-coordinate of the center is at \(-6\). Since there's no additional term next to \(y^2\), it indicates that the y-component comes from \(y - 0\), meaning the y-coordinate is \(0\). So, the center of the circle is \((-6, 0)\).Once you know this, you have automatically mapped out where the circle sits on the coordinate plane.

Identifying the radius from a circle's equation is straightforward once it's in standard form. In the form \((x - h)^2 + (y - k)^2 = r^2\),\( r\) represents the radius of the circle.For our equation \[(x+6)^2 + y^2 = 121\], the number \(121\)on the right side is equal to \(r^2\).This means we need to find the square root of \(121\) to determine the radius.

• Finding the square root of \(121\) is simple.It results in \(11\).

Therefore, the radius of the circle is \(11\).Whenever you're given a constant on the right side of these equations, taking the square root provides the circle’s radius. This is vital because it tells us how large the circle is and how far its edge is from the center.

The equation of a circle plays a central role in defining its properties and placement on the coordinate plane. It encapsulates two main factors: the center and the radius. The standard form for a circle’s equation is:\[(x - h)^2 + (y - k)^2 = r^2\]This form tells us:

• The coordinates \((h, k)\) which mark the center of the circle.
• The term \(r^2\) from which we can derive the radius.

To illustrate, let’s consider our example \((x+6)^2 + y^2 = 121\).By rearranging the terms, we identify the components that summarize the circle:- The \(x\)term’s structure \((x+6)transforming to (x - (-6))\)points to the center having an x-coordinate of \(-6\).- The lack of a y-term similar to \((y-k)\)means \(y\) is centered at \(0\).This highlights the circle's center at \((-6, 0)\),while- The right side, \(121, matches the form r^2\) indicating its radius is \(11\). Combining these insights gives us a circle centered at \((-6, 0)\), with a radius extending 11 units in all directions.