Question

If a, b, c, d and p are different real numbers such that (a² + b² + c²) p² – 2 (ab + bc + cd) p + (b² + c² + d²) £ 0, then a, b, c, d are in –

• A. A.P.
• B. G.P.
• C. H.P.
• D. ab = cd

Answer 1

Answer: (B)

G.P.

Answer 2

We have

(a² + b² + c²) p² – 2 (ab + bc + cd) p + (b² + c² + d²) £ 0 …(i)

LHS = (a²p² – 2abp + b²) + (b²p² – 2bcp + c²) +

(c²p² – 2cdp + d²)

= (ap – b)² + (bp – c)² + (cp – d)² ³ 0 …(ii)

Since, the sum of squares of real number is non-negative

From Equations (i) and (ii)

̃ (ap – b)² + (bp – c)² + (cp – d)² = 0

̃ ap – b = 0 = bp – c = cp – d

̃ $\frac { \mathrm { b } } { \mathrm { a } } = \frac { \mathrm { c } } { \mathrm { b } } = \frac { \mathrm { d } } { \mathrm { c } } = \mathrm { p }$

\ a, b, c, d are in G.P.