# Find the conditions on a and b such that line ax +by-1=0 intercepted bw lines ax+y+1=0 and x+by=0 subtend right angle at origin.

Akshay
185 Points
6 years ago
Find the intersection of ax+by=1 with 2 lines,
you will get two points, (1/(a-1) , -1/[b.(a-1)]) and (-(b+1)/[a.(b-1)] , 2/(b-1)),

To prove two point say A(x1,y1) and B(x2,y2) subtend angle $\Theta$ around origin, easiest way to check is using vectors,
Draw two vectors OA and OB,
OA = x1.i + y1.j,
OB = x2.i + y2.j,
Dot product, OA.OB = |OA|.|OB|.cos$\Theta$,
in this question $\Theta$=90, so dot product =0,
So, x1*x2 + y1*y2 = 0,
Putting x1,x2,y1,y2 you will get the answer: b(b+1) = -2a