Question

If b is the mean proportional between a and c, prove that `(a^2 - b^2 + c^2)/(a^-2 -b^-2 + c^-2)` = b⁴.

Answer 1

Since, b is the mean proportional between a and c. So, b² = ac.
L.H.S. = `(a^2 - b^2 + c^2)/(a^-2 -b^-2 + c^-2)`

= `(a^2 - b^2 + c^2)/(1/a^2 - 1/b^2 + 1/c^2)`

= `((a^2 - b^2 + c^2))/((b^2c^2 - a^2c^2 + a^2b^2)/(a^2b^2c^2)`

= `(a^2b^2c^2(a^2 - b^2 + c^2))/(b^2c^2 - b^4 + a^2b^2)`

= `(b^4 xx b^2 (a^2 - b^2 + c^2))/(b^2 (c^2 - b^2 + a^2)`
b⁴ = R.H.S.
Hence proved.