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Question

If p & q are distinct reals, then 2[(x−p)(x−q)+(p−x)(p−q)+(q−x)(q−p)]=(p−q)2+(x−p)2+(x−q)2 is satisfied for

A
no value of x
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B
exactly one value of x
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C
exactly two values of x
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D
infinite values of x
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Solution

The correct option is D infinite values of x
LHS = 2[(xp)(xq)+(px)(pq)+(qx)(qp)] = 2(x2xqpx+pq+p2pqpx+qx+q2pqqx+px) = 2(x2+p2+q2pxqxpq) = (2x2+2p2+2q22px2qx2pq)
RHS = (pq)2+(xp)2+(xq)2 = p2+q22pq+x2+p22px+x2+q22qx = (2x2+2p2+2q22px2qx2pq) = LHS
Since LHS=RHS,therefore all values of x satisfy the equation xR
Hence,(D) is the correct choice.

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