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)`

Answer 1

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.