Home
Class 12
MATHS
If 1/a+1/(a-b)+1/c+1/(c-b)= 0 and a + c-...

If `1/a+1/(a-b)+1/c+1/(c-b)= 0` and `a + c-b!=0`, then prove that `a, b, c` are in H.P.

Text Solution

Verified by Experts

We have, `(1)/(a)+(1)/(c)+(1)/(a-b)+(1)/(c-b)=0`
` implies ((1)/(a)+(1)/(c-b))+((1)/(c)+(1)/(a-b))=0`
` implies ((c-b+a))/(a(c-b))+((a-b+c))/(c(a-b))=0`
` implies (a+c-b)[(1)/(a(c-b))+(1)/(c(a-b))]=0`
` implies (a+c-b)[2ac-b(a+c)]=0`
If ` a+c-b ne 0," than "2ac-b(a+c)=0`
or `b=(2ac)/(a+c)`
Therefore,`a,b,c` are in HP and if `2ac-b(a+c) ne 0`, than `a+c-b=0 i.e., b=a+c`.
Promotional Banner

Similar Questions

Explore conceptually related problems

If a(1/b+1/c),b(1/c+1/a),c(1/a+1/b) are in A.P., prove that a,b,c are in A.P.

If a ,b ,c are in G.P. and a-b ,c-a ,a n db-c are in H.P., then prove that a+4b+c is equal to 0.

If (a+b)/(1-a b), b, (b+c)/(1-b c) are in A.P, then a, (1)/(b), c are in

If the lines a x+12 y+1=0,\ b x+13 y+1=0 and c x+14 y+1=0 are concurrent, then a , b , c are in

If a, b, c are in A.P., b, c, d are in G.P. and 1/c,1/d,1/e are in A.P. prove that a,c,e are in GP.

If a,b,c are in G.P and a^((1)/(x))=b^((1)/(y))=c^((1)/(z)) , prove that x,y,z are in A.P.

If 1/a,1/b,1/c are in A.P and a,b -2c, are in G.P where a,b,c are non-zero then

If the roots of the equations (b-c) x^(2) + (c-a) x+( a-b) =0 are equal , then prove that 2b=a+c

If a, b, c are positive real numbers such that a + b + c = 1 , then prove that a/(b + c)+b/(c+a) + c/(a+b) >= 3/2