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The equation A/(x-a1)+A2/(x-a2)+A3/(x-a3...

The equation `A/(x-a_1)+A_2/(x-a_2)+A_3/(x-a_3)=0 ,where A_1,A_2,A_3gt0 and a_1lta_2lta_3` has two real roots lying in the invervals.
(A) `(a_1,a_2) and (a_2,a_3)` (B) `(-oo,a_1) and (a_3,oo)` (C) `(A_1,A_3) and (A_2,A_3)` (D) none of these

Text Solution

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Points 1,2,3 and 4 are movable. Let their dis-placements from a fixed line be ` x_1 , x_2 , x_3 and x_4`
We have ` x_1 + x_4 = l_1` ( length of first string ) ..... (i)
and `(x_2 - x_4)+ (x_3 - x_4) = l_2` (length of second string )
or ` x_2 + x_3 - 2x_4 = l_2`............ (ii)
On double ,differentiating with respect to time ,
we get ` a_1 + a_4 =0` ............ (iii)
and ` a_2 + a_3 - 2a_4 = 0` ......... (iv)
But since ` a_4 = -a_1 a_4 = - a_1 ` [From equation (iii) ]
we have ,` a_2 + a_3 + 2a_1 =0`
This is the required constraint relation between ` a_1 , a_2 and a_3`
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