Name: Commensurable and Incommensurable Quantities
Uploaded: 2019-07-19
Duration: 6 min 35 s
Description: Two like quantities are known as commensurable if their ratio is a rational number. If the ratio of two like quantities is an irrational number, then they ar...

Question

If $b$ is the mean proportion between $a$ and $c$ , show that $\frac{a^{4} + a^{2} b^{2} + b^{4}}{b^{4} + b^{2} c^{2} + c^{4}} = \frac{a^{2}}{c^{2}} .$

Accepted Answer

Given that, $b$ is mean proportional between $a$ and $c$ i.e., $\frac{a}{b} = \frac{b}{c}$ .

Let, $\frac{a}{b} = \frac{b}{c} = x$

$\Rightarrow b = c x, a = b x = (c x) x = c x^{2}$

$L . H . S . = \frac{a^{4} + a^{2} b^{2} + b^{4}}{b^{4} + b^{2} c^{2} + c^{4}}$

$= \frac{(c x^{2})^{4} + (c x^{2})^{2} (c x)^{2} + (c x)^{4}}{(c x)^{4} + (c x)^{2} c^{2} + c^{4}}$

$= \frac{c^{4} x^{8} + c^{4} x^{6} + c^{4} x^{4}}{c^{4} x^{4} + c^{4} x^{2} + c^{4}}$

$= \frac{c^{4} x^{4} (x^{4} + x^{2} + 1)}{c^{4} (x^{4} + x^{2} + 1)}$ $= x^{4}$

Now,

$R . H . S . = \frac{a^{2}}{c^{2}} = \frac{(c x^{2})^{2}}{c^{2}} = \frac{c^{2} x^{4}}{c^{2}} = x^{4}$

Thus, $L . H . S . = R . H . S .$

Hence, $\frac{a^{4} + a^{2} b^{2} + b^{4}}{b^{4} + b^{2} c^{2} + c^{4}} = \frac{a^{2}}{c^{2}}$

If $b$ is the mean proportion between $a$ and $c$ , show that $\frac{a^{4} + a^{2} b^{2} + b^{4}}{b^{4} + b^{2} c^{2} + c^{4}} = \frac{a^{2}}{c^{2}} .$

Important Questions on Ratio and Proportion (Including Properties and Uses)

Learn and Prepare for any exam you want !

If b is the mean proportion between a and c, show that a4+a2b2+b4b4+b2c2+c4=a2c2.

Important Questions on Ratio and Proportion (Including Properties and Uses)

Learn and Prepare for any exam you want !

If $b$ is the mean proportion between $a$ and $c$ , show that $\frac{a^{4} + a^{2} b^{2} + b^{4}}{b^{4} + b^{2} c^{2} + c^{4}} = \frac{a^{2}}{c^{2}} .$