Write the standard form of the equation of the circle with the given center and radius. Center (0, 0), r = 7

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

Identify the center \((h, k)\) and radius \(r\) from the problem. Here, the center is \((0, 0)\) and the radius is \(7\).

Substitute the values of \(h = 0\), \(k = 0\), and \(r = 7\) into the standard form equation: \[ (x - 0)^2 + (y - 0)^2 = 7^2 \]

Simplify the equation by removing the zeros inside the parentheses and squaring the radius: \[ x^2 + y^2 = 49 \]

This equation represents the circle in standard form with the given center and radius.

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주요 개념

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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 clearly shows the circle's center coordinates and radius, making it easy to graph or analyze.

Center of the Circle

The center of a circle is the fixed point from which all points on the circle are equidistant. In the equation, the center is represented by (h, k). For this problem, the center is at the origin (0, 0), simplifying the equation.

Radius of the Circle

The radius is the distance from the center to any point on the circle. It is a positive value denoted by r in the equation. Squaring the radius (r²) is essential in the standard form to represent the circle's size accurately.