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Let a x+b y+c=0 be a variable straight l...

Let `a x+b y+c=0` be a variable straight line, whre `a , ba n dc` are the 1st, 3rd, and 7th terms of an increasing AP, respectively. Then prove that the variable straight line always passes through a fixed point. Find that point.

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Let the common difference of A.P.be d.
Then b=a+2d and c=a+6d.
So, given variable straight line will be
ax+(a+2d)y+a+6d=0
or a(x+y+1) + d(2y+6)=0
This is the equation of family of straight lines concurrent at point of intersection of lines x+y+1=0 and 2y+6=0 which (2,-3).
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