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If the roots of the eqn 1/(x+p) + 1/(x+p) = 1/r are equal in magnitude but opposite in sign, show that p+r=2r and that the product of the roots is equal to (-1/2) (p²+q²)

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I suspect a typo, since I get

2/(x+p ) = 1/r
x+p = 2r ---> linear equation
x = 2r-p , which only has one root

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