A line through `A(-5,-4)` meets the lines `x+3y+2=0,2x+y+4=0a n dx-y-5=0` at the points `B , Ca n dD` rspectively, if `((15)/(A B))^2+((10)/(A C))^2=(6/(A D))^2` find the equation of the line.

Explanation

Let equation of line `AC` is
`(y+4)/(sin theta)=(x+5)/(costheta)=r`
Let line `AE` make angle `theta` with `X`- axis and intersects `x+3y+2=0` at `B` at a distance `r_(1)` and line `2x+y+4=0` at `C` at a distance `r_(2)` and line `x-y-5=0` at `D` at a distance `r_(3)`.
`:.AB=r_(1)`, `AC=r_(2)`, `AD=r_(3)`.
`r_(1)=(-5-3xx4+2)/(1*costheta+3*sintheta)[:' r=-(I')/((alphacostheta+bsintheta))]`
`implies r_(1)=(15)/(costheta+3sintheta)`......`(i)`
Similarly, `r_(2)=-(2xx(-5)+1(-4)+4)/(2costheta+1*sintheta)`

`impliesr_(2)=(10)/(2costheta+sintheta)`.....`(ii)`
and `r_(3)=-(-5xx1-4(-1)-5)/(costheta-sintheta)`
`impliesr_(3)=(6)/(costheta-sintheta)`.........`(iii)`
But it is given that,
`((15)/(AB))^(2)+((10)/(AC))^(2)=((6)/(AD))^(2)`
`implies((15)/(r_(1)))^(2)+((10)/(r_(2)))^(2)=((6)/(r_(3)))^(2)`
`implies(costheta+3sintheta)^(2)+(2cos+sintheta)^(2)=(costheta-sintheta)^(2)` [from Eqs. `(i)`, `(ii)` and `(iii)`]
`impliescos^(2)theta+9sin^(2)theta+6costhetasintheta+4cos^(2)theta+sin^(2)theta+4costhetasintheta=cos^(2)theta+sin^(2)theta-2costhetasintheta`
`implies4cos^(2)theta+9sin^(2)theta+12sinthetacostheta=0`
`implies(2costheta+3sintheta)^(2)=0`
`implies2costheta+3sintheta=0`
`impliescostheta=-(3//2)sintheta`
Onsubsituting this in equation of `AC`, we get
`(y+4)/(sintheta)=(x+5)/(-(3)/(2)sintheta)`
`implies-3(y+4)=2(x+5)`
`implies-3y-12=2x+10`
`implies2x+3y+22=0`
which is the equation of required straight line.