Let A ( 0 , 1 ) , B ( 1 , 1 ) and C ( 1 , 0 ) be the mid-points of the sides of a triangle with incentre at the point D . If the focus of the parabola y 2 = 4 a x passing through D is ( α + β 2 , 0 ) , where α and β are rational numbers, then α β 2 is equal to

Step-by-step solution

Given, A ( 0 , 1 ) ,   B ( 1 , 1 ) and C ( 1 , 0 ) be the mid-points of the sides of a triangle with incentre at the point D , Now finding the vertices of the triangle by using midpoint formula and plotting the diagram we get, Now finding the incentre of the above triangle using the formula, I ≡ a x 1 + b x 2 + c x 3 a + b + c , a y 1 + b y 2 + c y 3 a + b + c We get, Incentre D = 4 4 + 2 2 , 4 4 + 2 2 Now given parabola y 2 = 4 a x passes through the incentre D we get, 4 4 + 2 2 2 = 4 a 4 4 + 2 2 &#8658