If a/b = c/d = e/f, prove that :(i) (b^2 + d^2 + f^2)(a^2 + c^2 + e^2) = (ab + cd + ef)^2(ii) a^3/b^3 + c^3/d^3 + e^3/f^3 = ace/(b d f)(iii) a^2/b^2 + c^2/d^2 + e^2/f^2 = ac/(bd) + ce/(df) + ae/(bf)(iv) (b d f) · [ (a + b)/b + (c + d)/d + (e + f)/f ]^3 = 27 (a + b)(c + d)(e + f)

Question

If a/b = c/d = e/f, prove that :(i)  (b^2 + d^2 + f^2)(a^2 + c^2 + e^2) = (ab + cd + ef)^2(ii) a^3/b^3 + c^3/d^3 + e^3/f^3 = ace/(b d f)(iii) a^2/b^2 + c^2/d^2 + e^2/f^2 = ac/(bd) + ce/(df) + ae/(bf)(iv) (b d f) · [ (a + b)/b + (c + d)/d + (e + f)/f ]^3 = 27 (a + b)(c + d)(e + f)

KnowledgeBoat · Accepted Answer

(i) Given, ⇒ (let) ⇒ a = kb, c = kd, e = kf. Substituting values of a,c and e in L.H.S. of (b2 + d2 + f2)(a2 + c2 + e2) = (ab + cd + ef)2, we get: Substituting values of a,c and e in R.H.S. of (b2 + d2 + f2)(a2 + c2 + e2) = (ab + cd + ef)2, we get: Since, L.H.S. = R.H.S. Hence, proved that (b2 + d2 + f2)(a2 + c2 + e2) = (ab + cd + ef)2 . (ii) Given, ⇒ (let) ⇒ a = kb, c = kd, e = kf. Substituting values of a,c and e in L.H.S. of , we get: SSubstituting values of a,c and e in R.H.S. of , we get: Since L.H.S. = R.H.S. Hence, proved that . (iii) Given, ⇒ (let) ⇒ a = kb, c = kd, e = kf. Substituting values of a,c and e in L.H.S. of , we get: Substituting values of a,c and e in R.H.S. of , we get: Since, L.H.S. = R.H.S. Hence, proved that . (iv) Given, ⇒ (let) ⇒ a = kb, c = kd, e = kf. Substituting values of a, c and d in L.H.S. of (bdf)·, we get: Substituting values of a, c and d in R.H.S. of (bdf)·, we get: Since, L.H.S. = R.H.S. Hence, proved that (bdf)..

Mathematics

If a/b = c/d = e/f, prove that :(i) (b^2 + d^2 + f^2)(a^2 + c^2 + e^2) = (ab + cd + ef)^2(ii) a^3/b^3 + c^3/d^3 + e^3/f^3 = ace/(b d f)(iii) a^2/b^2 + c^2/d^2 + e^2/f^2 = ac/(bd) + ce/(df) + ae/(bf)(iv) (b d f) · [ (a + b)/b + (c + d)/d + (e + f)/f ]^3 = 27 (a + b)(c + d)(e + f)

Ratio Proportion

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