General Form of a Circle

The equation x² +y² +2 gx +2 fy +c = 0 represents a circle iff g² +f² -c > 0.

Its center is (-g, -f) and radius = [g² +f² -c].

Concentric circles. Circles having same center are called concentric circles.

Equal circles Circles having equal radius are called equal circles.

Illustrative Examples

Example

One end of a diameter of the circle x² +y² -6 x +5 y -7 = 0 is (-1, 3). Find the co-ordinates of the other end.

Solution

The given equation is
   x² + y² -6 x +5 y -7 = 0
It is easy to see that this represents a circle with center C (3, - 5/2)
Let A(-1,3) and B(,) be the ends of the diameter.
Since C is mid-point of [AB], we get
   ( - 1)/2 = 3 and ( +3) / 2 = - 5/2
    = 7 and = -8
Hence the other end of the diameter is (7, -8).

Example

Show that the points (7, 5), (6, -2), (-1, -1) and (0, 6) are concyclic. Also find the radius and the center of the circle on which they lie.

Solution

Let us find the equation of the circle passing through the points(7, 5), (6, - 2) and (-1, -1).
Let the equation of this circle be
   x² + y² + 2gx + 2fy + c = 0            ...(i)
As the points (7, 5), (6, -2) and (-1, -1) lie on it, we get
49 +25 +14 g +10 f +c  => 14 g +10 f +c +74 = 0        ...(ii)
36 +4 +12 g -4 f +c = 0     =>      12 g -4 f + c +40 = 0     ...(iii)
1 -2 g -2 f +c = 0   =>  2 g +2 f -c -2 = 0          ...(iv)
Adding (ii) and (iv), we get
16 g +12 f +72 = 0   =>   4 g +3 f +18 = 0 ...(v)
Adding (iii) and (iv), we get
14 g -2 f +38 = 0      =>     7 g - f +19 = 0 ...(vi)
Solving (v) and (vi) simultaneously, we get g = -3, f = -2.     ...(vi)
From (ii), we get c = -14 (-3) -10 (-2) -74 = -12.
Substituting these values of g, f and c in (i), we get
x² +y² -6 x -4 y -12 = 0 ...(vii)
The fourth point (0, 6) will lie on (vii) if 0 +36 -0 -24 -12 = 0 i.e. if 0 = 0, which is true.
Hence the given points are concyclic.
Also, (vii) is the equation of the circle on which these points lie.
Its center is (3, 2) and radius = [9 +4 -(-12)] = 5.

Exercise

  1. Which of the following equations represent a circle? If so, determine its center and radius:
    (i) x² + y² +4 x -4 y -1 = 0
    (ii) 2 x² +2 y² = 3 x -5 y +7
    (iii) x² +y² +4 x +2 y +14 = 0
    (iv) 2 x² +2 y² = 5 x +7 y + 3
    (v) (x +3)² +(y -2)² = 0
    (vi) x² + y² - a x - b y = 0
  2. Find the value of p so that x² + y² +8 x; +10 y +p = 0 is the equation of a circle of radius 7 units.
  3. The radius of the circle x² +y² -2 x +3 y +k = 0 is 2. Find the value of k. Find also the equation of the diameter of the circle which passes through the point.
  4. (i) Find the equation of the circle the end points of whose one diameter are the centers of the circles x² +y² +6 x -14 y +5 = 0 and x² + y² -4 x +10 y +7 = 0.
    (ii) One end of a diameter of the circle x² + y² -3 x +5 y -4 = 0 is (2, 1), find the co-ordinates of the other end.
  5. Find the equation of the circle concentric with the circle x² +y² -8 x +6 y -5 = 0 and passing through the point (-2, -7).
  6. Find the equation of the circle which passes through the center of the circle x² +y² -4 x -8 y -41 = 0 and is concentric with the circle x² +y² -2 y +1 = 0.
  7. Find the equation of the circle concentric with the circle 2 x² +2 y² +8 x +10 y -35 = 0 and with area 16 square units.
  8. Find the equation of the circle which is concentric with the circle x² + y² -4 x +6 y -3 = 0 and of double its (i) circumference (ii) area.
  9. Prove that the centres of three circles x² + y² -4 x -6 y -14 = 0, x² + y² +2 x +4 y -5 = 0 and x² + y² -10 x -16 y +7 = 0 are collinear.
  10. Prove that the radii of the circles x² + y² = 1, x² + y² -2 x -6 y -6 = 0 and x² + y² -4 x -12 y -9 = 0 are in A.P.
  11. Find the equation of the circle which has its center on the line y = 2 and which passes through the points (2, 0) and (4, 0).
  12. Find the equation of the circle which passes through the points (1, - 2), (4, -3) and has its center on the line 3 x +4 y +10 = 0.
  13. Find the equation of the circle passing through the three points
    (i) (0, 0), (0, 1) and (2, 3)
    (ii) (0, 2), (3, 0) and (3, 2);
    Also find its center and radius.

Answers

1. (i) circle; (- 2 , 2), 3     (ii) circle; (3/4, 5/4), (310)/4     (iii) empty set
   (iv) circle; (5/4, 7/4), (v) point circle; (-3, 2), zero
     (vi) circle;(a/2, b/2)
2. -8                  3. -3; 2 x -2 y = 5
4. (i) x² + y² +x -2 y -41 = 0      (ii) (1, -6)
5. x² +y² -8 x +6 y -27 = 0
6. x² + y² -2 y -12 = 0
7. 4 x² +4 y² +16 x +20 y -23 = 0
8. (i) x² + y² -4 x +6 y -51 = 0    (ii) x² + y² -4 x +6y -19 = 0
11. x² +y² -6 x -4 y +8 = 0
12. x² +y² -4 x +8 y +15 = 0
13. (i) x² +y² -5 x - y = 0 -y = 0  ; (5/2, 1/2), 26 /2
      (ii) x² +y² -3 x -2 y = 0  ;  (3/2, 1), 13 /2