Consider a triangle having vertices A ( - 2 , 3 ) , B ( 1 , 9 ) and C ( 3 , 8 ) . If a line L passing through the circum-centre of triangle A B C , bisects line B C , and intersects y -axis at point 0 , α 2 , then the value of real number α is _______.

Step-by-step solution

We have, A B = - 2 - 1 2 + 3 - 9 2 = 45 A C = - 2 - 3 2 + 3 - 8 2 = 50 B C = 3 - 1 2 + 8 - 9 2 = 5 Here, A C 2 = A B 2 + B C 2 ⇒ ( 50 ) 2 = ( 45 ) 2 + ( 5 ) 2 Hence, ∠ B = 90 ° , so circumcentre is the mid-point of hypotenuse i.e., A C . Circumcenter ≡ 1 2 , 11 2 Mid-point of B C = 2 , 17 2 Equation of line L is y - 11 2 = 11 2 - 17 2 1 2 - 2 x - 1 2 ⇒ y = 2 x + 9 2 Passing though 0 , α 2 , so α 2 = 9 2 ⇒ α = 9