Question

if (a2 + c2), (ab + cd) and (b2 + d2) are in continued proportion. Prove that a,b,c and d are in proportion.

Answer 1

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Answer 2

a,b,c and d are in proportion proved below.

Step-by-step explanation:

Given:

$(a^2 + c^2), (ab + cd) and (b^2 + d^2)$ are in continued proportion.

⇒ $(a^2 + c^2) : (ab + cd) = (ab + cd) : (b^2 + d^2)$

⇒ $(a^2 + c^2) (b^2 + d^2) = (ab + cd) (ab + cd)$

⇒ $a^2b^2 + a^2d^2 + c^2b^2 + c^2d^2 = a^2b^2 + 2abcd + c^2d^2$

⇒ $a^2d^2 + c^2b^2 -2abcd =0$

⇒ $(ad -cb)^2 = 0$     [a^2d^2 + c^2b^2 -2abcd=(ad -cb)^2[/tex]]

⇒ ad - c = 0

⇒ ad = bc

⇒$\frac{a}{b} =\frac{c}{d}$

⇒ a, b, c and d are in proportion.