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 by

no value of

x

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exactly one value of

x

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exactly two values of

x

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infinite values of

x

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Solution

The correct option is D infinite values of $x$ .
$2 [(x - p) (x - q) + (p - x) (p - q) + (q - x) (q - p)] = (p - q)^{2} + (x - p)^{2} + (x - q)^{2}$
The above equation is a quadratic equation but it has three roots as $0$ , $p$ and $q$ satisfying it .
So, it is an identity and true for all $x$ .

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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 by

The correct option is D infinite values of x.2[(x−p)(x−q)+(p−x)(p−q)+(q−x)(q−p)]=(p−q)2+(x−p)2+(x−q)2The above equation is a quadratic equation but it has three roots as 0 , p and q satisfying it .So, it is an identity and true for all x .

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 by

The correct option is D infinite values of $x$ .
$2 [(x - p) (x - q) + (p - x) (p - q) + (q - x) (q - p)] = (p - q)^{2} + (x - p)^{2} + (x - q)^{2}$
The above equation is a quadratic equation but it has three roots as $0$ , $p$ and $q$ satisfying it .
So, it is an identity and true for all $x$ .