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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.

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.

Grade:12th pass

1 Answers

Akshay
185 Points
5 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

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