A line through the variable point A (k + 1, 2k) meets the lines 7x + y – 16 = 0, 5x – y – 8 = 0 , x – 5y + 8 = 0, at B, C, D then AC, AB, AD are in-
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Equation of line pass point A(k + 1, 2k) making angle θ with positive x-axis direction. Then
= 
= r 1 , r 2 , r 3 where AB = r 1 , AC = r 2 ,
AD = r 3
B ≡ ((k + 1) + r 1 cos θ , 2k + r 1 sin θ )
C ≡ ((k + 1) + r 2 cos θ , 2k + r 2 sin θ )
D ≡ ((k + 1) + r 3 cos θ , 2k + r 3 sin θ )
There points B, D, C satisfy limes 7x + y – 16 = 0, 5x – y – 8 = 0, x – 5y + 8 = 0 respectively
⇒ r 1 = 
, r 2 = 
and r 3 = 
= 
+ 
= 
= 
= 
⇒ AC, AB, AD are in H.P.
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