The line `3x+2y=24` meets the y-axis at `A` and the x-axis at `Bdot` The perpendicular bisector of `A B` meets the line through `(0,-1)` parallel to the x-axis at `Cdot` If the area of triangle `A B C` is `A` , then the value of `A/(13)` is________

Answer

91

Explanation

Lines 3x+2y = 24 meets the axes at B(8,0) and A(0, 12). The midpoint of AB is D(4, 6).
The equation of perpendicular bisector of AB is
`2x-3y+10=0 " " (1)`
Now, the line through (0, -1) and parallel to the x-axis is y=-1.
The coordinates of C where line (1) meets y =-1 are (-13/2, -1).
Now, the area of triangle ABC,
`Delta = (1)/(2)||{:(0, 12, 1),(8, 0, 1),(-13//2, -1, 1):}||`
`= (1)/(2)|[0-12(8+(13)/(2))+1(-8)]| = (1)/(2) |[-6(29)-8]| = 91`