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if (a^(2) +2bc) ,( b^(2) +2ac) ,(c^(2) ...

if ` (a^(2) +2bc) ,( b^(2) +2ac) ,(c^(2) +2ab) ` are in AP, show that
`1 / (( b-c)) ,1/((c -a)) , 1/ ((a-b)) are in AP.

Text Solution

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If `a^(2) + 2bc, b^(2) + 2ac, c^(2) + 2ab ` are in A.P.
Subtract ` ab + bc + ca `
` rArr (a^(2) + 2bc) - (ab + bc + ca ). (b^(2) + 2ac) - (ab + bc + ca), (c^(2) + 2ab) - (ab + bc + ca)` are in A.P.
`rArr a^(2) + bc - ab - ca , b^(2) + ca - ab - bc, c^(2)+ ab - bc - ca ` are in A.P.
`rArr (a - b)(a - c),(b - c)(b - a)(c - a)(c - b) ` are in A.P.
Divide by `(a - b) (b - c) (c - a)`
`rArr ((a - b) (a - c))/((a - b) (b - c)(c - a)),((b - c)(b - a))/((a - b)(b - c)(c - a)),((c - a)(c - b))/((a - b)(b -c)(c-a))` are in A.P.
`rArr (-1)/(b-c),(-1)/(c-a),(-1)/(a-b)` are in A.P. `rArr (1)/(b-c),(1)/(c-a),(1)/(a-b)` are in A.P.
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