Name: Proportion
Uploaded: 2019-07-19
Duration: 5 min 17 s
Description: A T-Rex looks strange because it has disproportionately sized arms. But, what are proportions? To understand their significance in mathematics, here are some...

Question

Find the mean proportional of ${(a^{4} - b^{4})}^{2}$ and $[(a^{2} + b^{2}) (a + b)]^{- 2}$ .

Accepted Answer

We have to find the mean of ${(a^{4} - b^{4})}^{2}$ and $[(a^{2} + b^{2}) (a + b)]^{- 2}$ .

A mean proportional is a number $x$ between two numbers $p$ and $q$ , for which $\frac{p}{x} = \frac{x}{q}$ .

Here, $p = {(a^{4} - b^{4})}^{2}$ and $q = [(a^{2} + b^{2}) (a + b)]^{- 2}$ .

Let $x$ is the mean between $p$ and $q .$

Therefore, $\frac{{(a^{4} - b^{4})}^{2}}{x} = \frac{x}{[(a^{2} + b^{2}) (a + b)]^{- 2}}$

$\Rightarrow x^{2} = {(a^{4} - b^{4})}^{2} \times [(a^{2} + b^{2}) (a + b)]^{- 2}$

$\Rightarrow x^{2} = \frac{{[{(a^{2})}^{2} - {(b^{2})}^{2}]}^{2}}{[(a^{2} + b^{2}) (a + b)]^{2}} = \frac{{(a^{2} + b^{2})}^{2} \times {(a^{2} - b^{2})}^{2}}{(a^{2} + b^{2})^{2} \times (a + b)^{2}}$

$\Rightarrow x^{2} = \frac{{(a^{2} - b^{2})}^{2}}{(a + b)^{2}} = \frac{{[(a + b) (a - b)]}^{2}}{(a + b)^{2}}$

$\Rightarrow x^{2} = \frac{{(a + b)}^{2} \times {(a - b)}^{2}}{(a + b)^{2}} = {(a - b)}^{2}$

$\Rightarrow x = (a - b)$

Hence, the required mean is $(a - b)$ .

Find the mean proportional of ${(a^{4} - b^{4})}^{2}$ and $[(a^{2} + b^{2}) (a + b)]^{- 2}$ .

Important Questions on Ratio and Proportion

Learn and Prepare for any exam you want !

Find the mean proportional of a4-b42 and [(a2+b2)(a+b)]-2.

Important Questions on Ratio and Proportion

Learn and Prepare for any exam you want !

Find the mean proportional of ${(a^{4} - b^{4})}^{2}$ and $[(a^{2} + b^{2}) (a + b)]^{- 2}$ .