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Question
If q is the mean proportional between p and r, prove that: p3 – 3q2 + r2 = `q^4(1/p^2 - 3/q^2 + 1/r^2)`
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Solution
q is mean proportional between p and r
q² = pr
L.H.S. = p2 – 3q2 + r2 = p2 – 3pr + r2
R.H.S. = `q^4(1/p^2 - 3/q^2 + 1/r^2)`
= `(q^2)^2(1/p^2 - 3/q^2 + 1/r^2)`
= `(pr)^2(1/p^2 - 3/q^2 + 1/r^2)`
= `(pr)^2(1/p^2 - 3/"pr" + 1/r^2)`
= `p^2r^2((r^2 - 3pr + p^2)/(p^2r^2))`
= r2 - 3pr + p2
∴ L.H.S. = R.H.S.
Hence proved.
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