Standard form of a circle :

 In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that

                                               (x - a)2 + (y - b)2 = r2

This equation of the circle follows from the Pythagorean Theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x − a and y − b. If the circle is centered at the origin (0, 0), then the equation simplifies to

                                                        x2 + y2 = r2

Source - Wikipedia

 

 

Example problems for Standard form of a circle:

 

1) Determine the standard form of circle equation with radius 9 and center (−5, 13).

Solution:

             Given:        r = 9     a = - 5    b = 13

          The standard form of a circle is given by

                               (x - a)2 + (y - b)2 = r2

                            (x - (-5))2 + (y - 13)2 = 92

                            (x + 5)2 + (y - 13)2  = 81

       This is the required standard form of the circle equation.

2) Determine the standard form of circle equation with radius 13 and center ( -4, 17).

Solution:

             Given:        r = 13     a = - 4    b = 17

          The standard form of a circle is given by

                               (x - a)2 + (y - b)2 = r2

                            (x - (-4))2 + (y - 17)2 = 132

                              (x + 4)2 + (y - 17)2  = 169

       This is the required standard form of the circle equation.

3) Determine the standard form of circle equation with radius 25 and center (−9, 20).

Solution:

             Given:        r = 25     a = - 9    b = 20

          The standard form of a circle is given by

                               (x - a)2 + (y - b)2 = r2

                            (x - (-9))2 + (y - 20)2 = 252

                             (x + 9)2 + (y - 20)2  = 625

This is the required standard form of the circle equation.

 

Practice questions for Standard form of circle:

1) Determine the standard form of circle equation with radius 6 and center (−3, 10).

Answer: (x + 3)2 + (y - 10)2  = 36

2) Determine the standard form of circle equation with radius 11 and center (−7, 24).

 Answer: (x + 7)2 + (y - 24)2  = 121