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# If a,b and C are real numbers and a2+b2+c2-ab-bc-ca =0 then show that a=b=c?

Asked by Vikas 20th March 2017, 5:58 PM

Consider,

a2 + b2 + c2 – ab – bc – ca = 0

Multiply both sides with 2, we get
2( a2 + b2 + c2 – ab – bc – ca) = 0
⇒ 2a+ 2b2 + 2c2 – 2ab – 2bc – 2ca = 0

⇒ a2 + a+ b2 + b+ c2 + c2– 2ab – 2bc – 2ca = 0
⇒ (a2 – 2ab + b2) + (b2 – 2bc + c2) + (c2 – 2ca + a2) = 0
⇒ (a – b)2 + (b – c)2 + (c – a)2 = 0
Since the sum of squares is zero, it means each term should be zero.
⇒ (a – b)2 = 0,  (b – c)2 = 0, (c – a)2 = 0
⇒ a – b = 0,  b – c = 0,  c – a = 0
⇒ a = b,  b = c, c = a
⇒ a = b = c

Answered by Expert 20th March 2017, 6:13 PM

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## From a point 36 m above the lake, the angle of elevation of a bird is observed to be 30 and the angle of depression of its image in water of the lake is observed to be 60. Find the actual height of the bird above the surface of the lake

Asked by Suneel 24th March 2018, 11:28 AM