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Question

If a+2b+3c=4 and k=a2+b2+c2, then the least possible value of 14k is

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Solution

Consider vectors p=a^i+b^j+c^k and q=^i+2^j+3^k
If θ is the angle between p and q, then
cosθ=a+2b+3ca2+b2+c212+22+32
cos2 θ=(a+2b+3c)214(a2+b2+c2)1
a2+b2+c287
Hence, the least value of a2+b2+c2 is 87.

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