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The equation of a circle is a crucial idea in coordinate geometry. It helps define the shape and location of a circle on the coordinate plane. The standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\). In this equation:

• \((h, k)\) represents the center of the circle.
• \(r\) is the radius, or the distance from the center to any point on the circle.

To understand better, compare the given circle equation \((x+3)^{2}+y^{2}=16\) with the standard form. You can see that it's structured similarly, which makes identifying the center and radius straightforward.
In the specific example, \((x+3)^2+y^2=16\), \(h\) is \(-3\), \(k\) is \(0\), and \(r^2=16\). This tells you the circle's center is shifted left by 3 units on the \(x\)-axis, and the radius squared is 16.

Determining the center and radius from a circle equation involves comparing it to the standard form and extracting values directly. The **center** of the circle is found at point \((h, k)\). In our example, \((x+3)^2 + y^2 = 16\) has

Finding the Center

The center, \(h\), is derived from the expression \((x+3)^2\), which corresponds to \((x-h)^2\). If you equate \((x-h)\) with \((x+3)\), \(h\) becomes \(-3\). Meanwhile, the \(y\) term \(y^2\) indicates that \(k\) is \(0\). Thus, the center is \((-3, 0)\).

Calculating the Radius

The radius, \(r\), is the square root of the equation's right side. From \(r^2=16\), the radius \(r\) calculates to \(4\). This value for the radius shows that all points on the circle are 4 units from the center in every direction.

Graphing a circle on the coordinate plane involves positioning its center and drawing its outline based on the radius.

Placing the Center

The center of the circle from our example is \((-3, 0)\). On a coordinate plane, this point is located 3 units left along the \(x\)-axis, while remaining at the origin level on the \(y\)-axis.

Drawing the Circle

Once centered at \((-3, 0)\), use the radius to draw the circle. Imagine or lightly draw a point 4 units outwards from the center in:

• The positive and negative directions along the \(x\)-axis and \(y\)-axis, creating points like \((1,0)\) and \((-3,4)\).
• These points guide you to sketch a round shape.

By ensuring all these steps are followed, you map out a perfect circle that reflects the mathematical equation accurately on your plane.