A line L through A(-5,-4) meets the lines x+3 y+2=0 and 2 x+y+4=0 , x-y-5=0 at points B, C and D respectively. If ( 15 A B )^2+ ( 10 A C )^2= ( 6 A D )^2 , then find the equation of L

Step-by-step solution

Let equation of line passing through A(-5,-4) is given by x-(-5) = y-(-4) =r x+5 = y+4 =r Therefore, every point on the line is of the form x=-5+r , y=-4+r Let A B=r₁, A C=r₂, A D=r₃ aligned & B= (-5+r₁ ,-4+r₁ ) and lies on & x+3 y+2=0 & -5+r₁ +3 (-4+r₁ )+2=0 & r₁= 15 +3 & 15 r₁ = +3 ...(i) aligned C= (-5+r₂ ,-4+r₂ ) and lies on 2 x+y+4=0 2 (-5+r₂ )+ (-4+r₂ )+4=0 10 r₂ =2 + ...(ii) D= (-5+r₃ ,-4+r₃ ) and lies on x-y-5=0 aligned & -5+r₃ - (-4+r₃ )-5=0 & 6 r₃ = - ...(iii) aligned aligned & Given that, ( 15 A B )^2+ (