Let the abscissae of the two points P and Q on a circle be the roots of x 2 - 4 x - 6 = 0 and the ordinates of P and Q be the roots of y 2 + 2 y - 7 = 0 . If P Q is a diameter of the circle x 2 + y 2 + 2 a x + 2 b y + c = 0 , then the value of a + b - c is

Step-by-step solution

Given that the roots of x 2 - 4 x - 6 = 0 are the abscissa of the end of diameter i.e. x 1 + x 2 = 4 ,   x 1 x 2 = - 6 and roots of y 2 + 2 y - 7 = 0 are ordinate of the end of diameter i.e. y 1 + y 2 = - 2 ,   y 1 y 2 = - 7 Now, equation of the circle will be x - x 1 x - x 2 + y - y 1 y - y 2 = 0 i.e. x 2 - x 1 + x 2 x + x 1 x 2 + y 2 - y 1 + y 2 y + y 1 y 2 = 0 ⇒ x 2 + y 2 - 4 x + 2 y - 13 = 0 ∴ a = - 2 ,    b = 1 ,    c = - 13 ⇒ a + b - c = - 2 + 1 + 13 = 12