. . Equation of a circle in standard form (i) Equation of circle with centre ( , ) and radius r Let the centre be C ( , ) and radius be r and P x y ( , ) be the moving point. Note that the point P having coordinates ( , ) x y is represented as P x y ( , ) .
Then, CP = r and so CP Therfore ( = r That is x = r This is the equation of the circle with centre ( , ) and radius r . Fig. . Fig.
. P ( x, y ) C ( , ) - - (ii) Equation of circle with centre ( h, k ) and radius r Let the centre be C h k ( , ) and r be the radius and P x y ( , ) be the moving point. Then, CP and so CP That is, ( h .This is the equation of the circle in Standard form, which is also known as centre-radius form. Expanding the equation, we get hx ky h Taking g h f k c h , the equation takes the form gx fy , called the general form of a circle.
The equation x gx fy is a second degree equation in x and y possessing the following characteristics: (i) It is a second degree equation in x and y , (ii) coefficient of x = coefficient of y ¹ (iii) coefficient of xy = . Conversely, we prove that an equation possessing these three characteristics, always represents a circle. Let ax ay g x f y ′ + ′ + … ( ) be a second degree equation in x and y having characteristics (i), (ii) and (iii), a ¹ . Dividing ( ) by a , gives g a x f a y ′ ′ + ′ = .
… ( ) Taking ′ = ′ = g g f f and ′ = c , equation ( ) becomes x gx fy = . Adding and subtracting g and f , we get x gx g fy f g f ⇒ ( g f g f ⇒ ( )) )) g f g f −− −− This is in the standard form of a circle with centre C ( g f and radius r = g f . Hence equation ( ) represents a circle with centre ( −′ −′ g f g f and radius ′ + ′ −′ g f g f c a Note The equation of the circle x gx fy with centre (– g , – f ) and radius g f − represents. (i) a real circle if g f > ; (ii) a point circle if g f ; (iii) an imaginary circle if g f < with no locus.
P ( x, y ) C ( h, k ) Fig. . - - Two Dimensional Analytical Geometry - II Example . Find the general equation of a circle with centre( and radius 3units.
Equation of the circle in standard form is x h ) + ) =