Write the standard form of the equation of the circle with the given center and radius. Center (-4, 0), r = 10

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Recall the standard form of the equation of a circle is given by: \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius.

Identify the center \( (h, k) \) from the problem: here, the center is \( (-4, 0) \), so \( h = -4 \) and \( k = 0 \).

Substitute the values of \( h \) and \( k \) into the standard form: \( (x - (-4))^2 + (y - 0)^2 = r^2 \).

Simplify the expression inside the parentheses: \( (x + 4)^2 + y^2 = r^2 \).

Substitute the radius \( r = 10 \) and square it to get \( r^2 = 10^2 = 100 \), so the equation becomes \( (x + 4)^2 + y^2 = 100 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Form of a Circle's Equation

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form directly relates the coordinates of any point on the circle to its center and radius.

Coordinates of the Center

The center of the circle is given as a point (h, k). In the equation, these values are used as (x - h) and (y - k) to measure the horizontal and vertical distances from the center to any point on the circle.

Radius and Squaring

The radius r is the distance from the center to any point on the circle. In the equation, the radius is squared (r²) to relate it to the sum of the squared distances along the x and y axes, ensuring all points satisfy the circle's definition.

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