Home
Class 12
MATHS
If a,b,c are in A.P. and a^(2) , b^(2) ,...

If a,b,c are in A.P. and `a^(2) , b^(2) , c^(2) ` are in H.P. , then `:`

Text Solution

Verified by Experts

Given, `a,b,c` are in AP.
`therefore " " b=(a+c)/(2) " " "……(i)`
and `a^(2),b^(2),c^(2)` in HP.
`therefore " " b^(2)=(2a^(2)c^(2))/(a^(2)+c^(2)) " " "………(ii)`
From Eq. (ii) `b^(2){(a+c)^(2)-2ac}=2a^(2)c^(2)`
` implies b^(2){(2b)^(2)-2ac}=2a^(2)c^(2) " " [" from Eq. (i)"]`
`implies 2b^(4)-acb^(2)-a^(2)c^(2)=0`
`implies (2b^(2)+ac)(b^(2)-ac)=0`
`implies 2b^(2)+ac =0` or `b^(2)-ac=0`
If `2b^(2)+ac=0`, than `b^(2)=-(1)/(2)ac` or `-(a)/(2),b,c` are in GP
and if `b^(2)-ac =0 implies a,b,c` are in GP.
But given, `a,b,c` are in AP.
Which is possible only when `a=b=c`.
Promotional Banner

Similar Questions

Explore conceptually related problems

Suppose a,b,c are in A.P. and a^(2) , b^(2) , c^(2) are in G.P. If a lt b lt c and a+ b + c = ( 3)/( 2) , then the value of a is :

If a,b,c are in A.P., the 3^(a), 3^(b) , 3^(c ) are in :

If a,b,c are in A.P., then 7^(a) , 7^(b) and 7^(c ) are in :

If a,b,c,d are in H.P., then :

Suppose a,b,c are in AP and a^(2),b^(2),c^(2) are in GP, If agtbgtc and a+b+c=(3)/(2) , than find the values of a and c.

If a,b,c are in A.P. and a,mb,c are in G.P. , then a, m^(2)b, are in :

In a triangle A B C, if cot A, cot B, cot C are in A.P, then a^(2), b^(2), c^(2) are in

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, c are in A . P , then (a-c)^(2)=

If a,b,c are three unequal numbers such that a,b,c are in A.P. And b-a, c -b , a are in G.P., then a: b : c is :