Question

If a, b, c and d are in proportion, prove that: `(a^2 + b^2)/(c^2 + d^2) = "ab + ad - bc"/"bc + cd - ad"`

Answer 1

∵ a, b, c, d are in proportion
`a/b = c/d` = k(say)
a = bk, c = dk.
L.H.S. = `(a^2 + b^2)/(c^2 + d^2)`

= `(b^2k^2 + b^2)/(d^2k^2 + d^2)`

= `(b^2(k^2 + 1))/(d^2(k^2 + 1)`

= `b^2/d^2`

R.H.S. = `"ab + ad - bc"/"bc + cd - ad"`

= `"bk.b + bk.d - b.dk"/"b.kd + dk.d - bk.d"`

= `(k(b^2 + bd - bd))/(k(bd + d^2 - bd)`

= `b^2/d^2`
∴ L.H.S. = R.H.S.