Question

[Maths]
Topic: Ratio and Proportion

$\sf \: print("Hlo \: Brainlians!!")$
Help!!!

If b is the mean proportion between a and c, Prove:
$\: \: \: \: \underline{\boxed{ \boxed{ \bold{ \orange{ \dfrac{ {a}^{2} - {b}^{2} + {c}^{2} }{ {a}^{ - 2}- {b}^{ - 2} + {c}^{ - 2} } = {b}^{4} }}}}}$
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Answer 1

$\texttt{\textsf{\large{\underline{Solution!}}}}$

Given:

→ b is the mean proporrtion between a and c.

To Prove:

$\tt\implies\dfrac{a^{2}-b^{2}+c^{2}}{a^{-2}-b^{-2}+c^{-2}} = b^{4}$

As b is the mean proportion between a and c:

$\tt\implies \dfrac{a}{b}=\dfrac{b}{c}$

$\tt\implies b^{2}=ac$

Taking LHS:

$\tt=\dfrac{a^{2}-b^{2}+c^{2}}{a^{-2}-b^{-2}+c^{-2}}$

$\tt=\dfrac{a^{2}-b^{2}+c^{2}}{\dfrac{1}{a^{2}}-\dfrac{1}{b^{2}}+\dfrac{1}{c^{2}}}$

$\tt=\dfrac{a^{2}-b^{2}+c^{2}}{\dfrac{b^{2}c^{2}-a^{2}c^{2}+a^{2}b^{2}}{a^{2}b^{2}c^{2}}}$

$\tt=\dfrac{a^{2}b^{2}c^{2}(a^{2}-b^{2}+c^{2})}{b^{2}c^{2}-a^{2}c^{2}+a^{2}b^{2}}$

$\tt=\dfrac{(ac)^{2}b^{2}(a^{2}-b^{2}+c^{2})}{b^{2}c^{2}-(ac)^{2}+a^{2}b^{2}}$

Put b² = ac:

$\tt=\dfrac{b^{4}\cdot b^{2}(a^{2}-b^{2}+c^{2})}{b^{2}c^{2}-b^{4}+a^{2}b^{2}}$

$\tt=\dfrac{b^{4}\cdot b^{2}(a^{2}-b^{2}+c^{2})}{b^{2}(c^{2}-b^{2}+a^{2})}$

$\tt=\dfrac{b^{4}(a^{2}-b^{2}+c^{2})}{c^{2}-b^{2}+a^{2}}$

$\tt=b^{4}$

= RHS!

Hence, Proved.

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Properties of Ratio and Proportion.

1. Invertendo: If a : b : : c : d then b : a : : d : c

2. Alternendo: If a : b : : c : d then a : c : : b : d

3. Componendo: If a : b : : c : d then (a + b) : b : : (c + d) : d

4. Dividendo: If a : b : : c : d then (a - b) : b : : (c - d) : d

5. Componendo and Dividendo: If a : b : : c : d then (a + b) : (a - b) : : (c + d) : (c - d)

6. Convertendo: If a : b : : c : d then a : (a - b) : : c : (c - d)