Question

If q is the mean proportional between p and r, prove that
`p^2 - q^2 + r^2 = q^4[(1)/p^2 - (1)/q^2 + (1)/r^2]`.

Answer 1

Since, q is the mean proportional of p and r.
Hence, q2 = pr.
R.H.S. = `q^4[(1)/p^2 - (1)/q^2 + (1)/r^2]`
= `q^4[(1)/p^2 - (1)/(pr) + (1)/r^2]`
= `q^4[(r^2 - pr + p^2)/(p^2r^2)]`
= `q^4[(p^2 - pr + r^2)/(pr)^2]`
= `q^4[(p^2 - pr + r^2)/q^4]`
= p² - pr + r²
= p² - q² + r² = L.H.S.
Hence proved.