Question

If `a/c = c/d = c/f` prove that : `(a^2)/(b^2) + (c^2)/(d^2) + (e^2)/(f^2) = "ac"/"bd" + "ce"/"df" + "ae"/"df"`

Answer 1

`a/c = c/d = c/f` = k(say)
∴ a = bk, c = dk, e =fk
L.H.S. = `(a^2)/(b^2) + (c^2)/(d^2) + (e^2)/(f^2)`

= `(b^2k^2)/(b^2) + (d^2k^2)/(d^2) + (f^2k^2)/(f^2)`
= k² + k² + k²
= 3k²
R.H.S. = `"ac"/"bd" + "ce"/"df" + "ae"/"bf"`

= `"bk.dk"/"b.d" + "dk.fk"/"d.f" + "bk.fk"/"b.f"`
= k² + k² + k²
= 3k² 
∴ L.H.S. = R.H.S.