Let a,b,c,d and p be any non zero distinct real numbers such that (a2

1.	Let a,b,c,d and p be any non zero distinct real numbers such that (a2 + b2 + c2)p2 – 2(ab + bc + cd)p + (b2 + c2 + d2) = 0. Then : (1) a,c,p are in G.P. (2) a,c,p are in A.P. (3) a,b,c,d are in G.P. (4) a,b,c,d are in A.P.
Answer» (3) a,b,c,d are in G.P. (a² + b² + c²)p² + 2(ab + bc + cd)p + b² + c² + d² = 0 (a²p² + 2abp + b²) + (b²p² + 2bcp + c²) + (c²p² + 2cdp + d²) = 0 (ab + b)² + (bp + c)² + (cp + d)² = 0 This is possible only when ap + b = 0 and bp + c = 0 and cp + d = 0 p = -b/a = -c/b = -d/c or b/a = c/b = d/c ∴ a,b,c,d are in G.P.

Let a,b,c,d and p be any non zero distinct real numbers such that (a2 + b2 + c2)p2 – 2(ab + bc + cd)p + (b2 + c2 + d2) = 0. Then : (1) a,c,p are in G.P. (2) a,c,p are in A.P. (3) a,b,c,d are in G.P. (4) a,b,c,d are in A.P.

Answer»

(3) a,b,c,d are in G.P.

(a² + b² + c²)p² + 2(ab + bc + cd)p + b² + c² + d² = 0

(a²p² + 2abp + b²) + (b²p² + 2bcp + c²) + (c²p² + 2cdp + d²) = 0

(ab + b)² + (bp + c)² + (cp + d)² = 0

This is possible only when

ap + b = 0 and bp + c = 0 and cp + d = 0

p = -b/a = -c/b = -d/c

or b/a = c/b = d/c

∴ a,b,c,d are in G.P.

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Let a,b,c,d and p be any non zero distinct real numbers such that (a2 + b2 + c2)p2 – 2(ab + bc + cd)p + (b2 + c2 + d2) = 0. Then : (1) a,c,p are in G.P. (2) a,c,p are in A.P. (3) a,b,c,d are in G.P. (4) a,b,c,d are in A.P.

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