A) \[83x-35y+92=0\]
B) \[35x-83y+92=0\]
C) \[35x+35y+92=0\]
D) None of these
Correct Answer: A
Solution :
Any line through the middle point M(1, 5) of the intercept AB may be taken as \[\frac{x-1}{\cos \theta }=\frac{y-5}{\sin \theta }=r\] ?..(i) where ?r? is the distance of any point (x, y) on the line (i) from the point M(1, 5). Since the points A and B are equidistant from M and on the opposite sides of it, therefore if the coordinates of A are obtained by putting r=d in (i), then the co-ordinates of B are given by putting \[r=-d\]. Now the point \[A(1+d\cos \theta ,\,5+d\sin \theta )\]lies on the line \[5x-y-4=0\] and point \[B(1-d\cos \theta ,\,5-d\sin \theta )\] lies on the line \[3x+4y-4=0\]. Therefore, \[5(1+d\cos \theta )-(5+d\sin \theta )-4=0\] and \[3(1-d\cos \theta )+\]\[4(5-d\sin \theta )-4=0\] Eliminating ?d? from the two, we get \[\frac{\cos \theta }{35}=\frac{\sin \theta }{83}\]. Hence the required line is \[\frac{x-1}{35}=\frac{y-5}{83}\] or \[83x-35y+92=0\].
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