A line `L_1=3y-2x-6=0` is rotated about its point of intersection with the y-axis in the clockwise direction to make it `L_2` such that the are formed by `L_1,L_2` the x-axis, and line `x=5` is `(49)/3s qdotu n i t s` if its point of intersection with `x=5` lies below the x-axis. Find the equation of `L_2dot`

Answer

x+y=2

Explanation

`L_(1)=3y-2x-6=0`
The point about which the line is rotated is A = (0,2). Let the equation of `L_(2)` be y = mx +2. As `L_(2)` will be cutting line x=5 below the x-axis, we have
`A-=(0,2),B-=(5,(16)/(3)), C-= (5,0),`
`D-=(5,5m +2),E-=((-2)/(m),0)`
`"Now, Area of AECB" =" Area of "Delta ADB-"Area of " Delta ECD`
`" or " (49)/(3) = (1)/(2)BD xx AF-(1)/(2)EC xx CD`
`=(1)/(2)[(16)/(3)-(5m+2)] xx 5-(1)/(2) xx(5+(2)/(m))[-(5m+2)]`
`" or " (49)/(3) = (110m+12)/(6m) "or m" = -1`
`"Hence, the equation of "L_(2) " is " x+y=2.`