If 2 x + 3 y + 4 = 0 is the perpendicular bisector of the line segment joining the points A ( 1 , 2 ) and B ( α , β ) , then the value of 13 α + 13 β equals_____

Step-by-step solution

Here, Since the line is bisector of the line segment joining the points, The distance of B from the line will be equal to the distance of A from the line. Hence, 2 + 6 + 4 13 = 2 α + 3 β + 4 13 ⇒ 2 α + 3 β + 4 = 12 ⇒ 2 α + 3 β + 4 = ± 12 ⇒ 2 α + 3 β = 8 and 2 α + 3 β = - 16     . . . . ( i ) Also, the line joining the points is perpendicular to 2 x + 3 y + 4 = 0 , Thus the product of their slopes is - 1 . ⇒ β - 2 α -