5.1 Circles 1 The following are several definitions necessary for the understanding of circles. 1.) Circle - A set of points equidistant from a given fixed point on a plane. 2.) Center - The point from which all other points of a circle are equidistant from. 3.) Radius - The distance from the center of a circle to its edge. (r=radius) ● r ● x y 4.) Diameter - The distance from one edge of a circle to the other side through the center. (d=diameter) ● ● d ● x y ● r ● x y P(x,y) C(h,k) The formula for the radius is r =, where (h, k) represents the center of the circle and (x, y) is a point on the edge. When both sides of this equation are squared the result is the standard form equation of a circle: Performing the exponentiation and simplifying the equation by getting all of the terms on the same side will give you another form in which the equation of a circle can be expressed. This is called the general form of a circle.

5.1 Circles 2 Procedure: Finding the equation of a circle in general form given the center and radius. 1. Replace h, k and r with the values given into the standard form of a circle. 2. Simplify by performing the exponentiation and getting all of the terms on the same side. Example 1.Find the equation of a circle, in general form, having a center at (-4,5) and a radius of length 3 units. Step 1.Substitute the values for h, k and r into the standard formula. Step 2.Perform the exponentiation and get all terms on the same side. Your Turn Problem #1 Find the equation of a circle, in general form, having a center at (2,-6) and a radius of length 4 units.

5.1 Circles 3 Procedure:Finding the equation of a circle in general form given the center and a point on the circle. 1. Find the radius using the formula given, i.e. the distance formula. 2. Substitute h, k and r into the standard form of a circle, then simplify by performing the exponentiation and getting all of the terms on the same side. Next Slide To find an equation of a circle in either form, we need the center and the radius. If a point on a circle is given along with the center, we can use the formula r = to find the radius. Note: This formula was derived by using the distance formula:

5.1 Circles 4 Example 2.Find the equation of a circle, in general form, having a center at (3,-2) and passing through (-1,1) on its edge. Step 1.Find the radius by substituting the values of the points into the radius formula. Step 2.Find the general form of the equation. h k (3,-2) x y (-1,1) Your Turn Problem #2 Find the equation of a circle, in general form, having a center at (  4,5) and passes through the point (3,  4). Note: The radius could also be found by replacing h and k with (3,-2) and replacing x and y with (-1,1).

5.1 Circles 5 Procedure: To Find the standard form of a circle given the general form of a circle. 1.Move the constant to the right hand side and rearrange the terms as follows: (x 2 + Dx + __) + (y 2 + Ey + ___) = -F. The coefficients of the x 2 and y 2 must be 1. The goal in the previous examples was to obtain the general form, x 2 + y 2 + Dx +Ey + F = 0, given the center and radius or the center and a point on the circle. In the next two examples, the general form will be given and the goal will be to obtain the standard form of a circle, the center and length of the radius. 2.Then create two perfect square trinomials using the technique of completing the square to obtain the standard form: (x – h) 2 + (y – k) 2 = r 2. Example 3.Write the equation in standard form and state the center and radius. 1.Move the constant to the right hand side and rearrange the terms with parentheses. 2.Use the technique of completing the square to write in standard form. Your Turn Problem #3 Write the given equation of the circle in standard form and state the center and radius.

5.1 Circles 6 Step 1.Use the technique of completing the square. Example 4.Write the equation of the circle in standard form and state the center and radius. Note: To find the radius, take the square root of 19/2. Then rationalize the denominator. Your Turn Problem #4 Write the equation of the circle in standard form and state the center and radius. The End B.R