A circle in the standard \((x, y)\) coordinate plane is tangent to the \(x\) -axis at 5 and tangent to the \(y\) -axis at \(5 .\) Which of the following is an equation of the circle? A. \(x^{2}+y^{2}=5\) B. \(x^{2}+y^{2}=25\) C. \((x-5)^{2}+(y-5)^{2}=5\) D. \((x-5)^{2}+(y-5)^{2}=25\) E. \((x+5)^{2}+(y+5)^{2}=25\)

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This method combines algebra and Euclidean geometry, allowing problems to be solved in a way that is not possible with synthetic geometry alone.

For circles, coordinate geometry is particularly useful, as the locus of points that make up a circle can be defined by a specific equation. The standard form of the equation of a circle in the xy-plane is \[ (x-a)^{2} + (y-b)^{2} = r^{2} \], where \(a, b)\ is the center of the circle, and \(r\) is the radius. Any point \( (x, y) \) that satisfies this equation lies on the circle.

In the context of our problem, the circle is tangent to both the x-axis and y-axis, indicating that its center is positioned at an equal distance from both axes. Therefore, using the coordinates of the tangency points, it's possible to determine the center of the circle, which is a crucial step in formulating the equation of the circle within a coordinate plane.

In geometry, a tangent to a circle is a straight line that touches the circle at exactly one point, known as the point of tangency. This property is crucial in solving problems related to circles and tangents.

An important property to remember is that a tangent to a circle at a point is perpendicular to the radius at that point. This means that if a line is tangent to a circle at point P, and the center of the circle is point O, then the radius OP is perpendicular to the tangent line at P.

In our exercise, the circle is tangent to the x-axis at \(5,0\) and the y-axis at \(0,5\). This gives us valuable information about the position of the circle relative to these axes. Because the tangents from a point outside the circle are equal in length, the distance from the center of the circle to these tangency points is the same, leading us directly to the calculation of the circle's radius.

Calculating the radius of a circle is straightforward when we know the coordinates of its center and a point on the circle. Since we are dealing with a circle tangent to both the x-axis and y-axis, the radius can be found by measuring the distance from the center of the circle to the point of tangency.

The formula to calculate the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the xy-plane is given by \[ R =\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].

In the solution to our exercise, we used the distance formula to find the radius, with the center of the circle being at \( (5,5) \) and a tangency point at \( (5,0) \). This calculation yielded a radius of 5 units, allowing us subsequently to write out the full equation of the circle. Ensuring the proper calculation of the circle's radius is essential as it affects the entire equation and ultimately the solution to the problem.