Show that the lines `4x+y-9=0,x-2y+3=0,5x-y-6=0` make equal intercepts on any line of slope 2.

Explanation

The given lines are
` 4x+y-9= 0 " "(1)`
`x-2y+3=0 " " (2)`
`5x-y-6=0 " " (3)`
Now, the equation of any line having slope (gradient) 2 will be
`y=2x+c " "(4)`
Let line (4) cuts line (1), (2), and (3) at A,B, and C. respectively.
Solving (1) and (4), we get
`A -=((3)/(2) -(c)/(6),3+(2c)/(3))`
`"Similary,"B -=(1-(2c)/(c),2-(c)/(3))`
`"and " C -=(2+(c)/(3),4+(5c)/(3))`
Clearly , A is the middle point of BC, Hene, AB=AC.