The Straight Line: Equation of a Circle

Understanding the Definition of a Circle

A circle is a set of points in a plane equidistant from a fixed point called the center.
If (h, k) are the coordinates of the center of the circle and r is the radius, then the equation of the circle will be: (x - h)² + (y - k)² = r².

To identify the center (h, k) and the radius r from the equation of the circle (x - h)² + (y - k)² = r², remember to note that h and k are the x and y coordinates of the center, and r is the radius.
If the equation of the circle is given in general form (x² + y² + 2gx + 2fy + c = 0), you can rewrite it in standard form to find h and k. Here h = -g and k = -f, and r = sqrt(g² + f² - c).

To plot a circle on a graph, identify the center and radius from its equation.
Mark the center of the circle on the graph at the point (h, k).
From the center, draw a circle with the identified radius r by measuring the radius along both axes from the center point.

When given a task to find the equation of a circle from given points, use the distance formula, which is derived from Pythagoras’ theorem, to find the radius (r) and use the given center (h, k).
If the radius and a point on the circle are given, substitute these into the equation (x - h)² + (y - k)² = r² to find the equation.
If two points are given (one being the center and the other lying on the circle), calculate the distance between the points to get the radius.

Remember, the key to mastering the equation of a circle is practice. Solve different problems based on the equation of a circle to gain confidence.