and the radius of the circle x + y + x + y – = Solution The given equation is ( x + x ) + ( y + y ) = Now, completing the squares within the parenthesis, we get ( x + x + ) + ( y + y + ) = + + i.e. ( x + ) + ( y + ) = i.e. { x – (– )} + { y – (– )} = Therefore, the given circle has centre at (– , – ) and radius . Fig .
Fig . CONIC SECTIONS Example Find the equation of the circle which passes through the points ( , – ), and ( , ) and whose centre lies on the line x + y = . Solution Let the equation of the circle be ( x – h ) + ( y – k ) = r . Since the circle passes through ( , – ) and ( , ), we have ( – h ) + (– – k ) = r ...
( ) and ( – h ) + ( – k ) = r ... ( ) Also since the centre lies on the line x + y = , we have h + k = ... ( ) Solving the equations ( ), ( ) and ( ), we get h = . , k = .
and r = . Hence, the equation of the required circle is ( x – . ) + ( y – . ) = .
. EXERCISE . In each of the following Exercises to , find the equation of the circle with . centre ( , ) and radius .
centre (– , ) and radius . centre ( , ) and radius . centre ( , ) and radius . centre (– a , – b ) and radius b In each of the following Exercises